Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 918513 7 pageshttpdxdoiorg1011552013918513

Research ArticleAsymptotic Estimates for Second Kind GeneralizedStirling Numbers

Cristina B Corcino1 and Roberto B Corcino2

1 Institute of Mathematics University of the Philippines Diliman 1101 Quezon City Philippines2 Department of Mathematics Mindanao State University Main Campus 9700 Marawi City Philippines

Correspondence should be addressed to Roberto B Corcino rcorcinoyahoocom

Received 5 June 2013 Accepted 1 August 2013

Academic Editor Zhijun Liu

Copyright copy 2013 C B Corcino and R B Corcino This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained andranges of validity of the formulas are establishedThe generalizations of Stirling numbers considered here are generalizations alongthe line of Hsu and Shuiersquos unified generalization

1 Introduction

Asymptotic formulas of the classical Stirling numbers havebeen done by many authors like Temme [1] Moser andWyman [2 3] due to the importance of the formulas incomputing values of the numbers under consideration whenparameters become large The Stirling numbers and theirgeneralization on the other hand are important due to theirapplications in statistics life science and physics

The (119903 120573)-Stirling numbers [4] the 119903-Whitney numbersof the second kind [5] and the numbers considered byRucinski and Voigt [6] are exactly the same numbers whichcan be classified as generalization of the classical Stirlingnumbers of the second kindThis generalization is in linewiththe generalization of Hsu and Shuie [7] For brevity we canuse 119878120573119903(119899119898) to denote these numbersThese numbers satisfythe following exponential generating function

1

120573119898 (119898)119890119903119911(119890120573119911minus 1)119898=

infin

sum119899=119898

119878120573119903 (119899119898)119911119899

119899 (1)

where 119899 and119898 are positive integers When 120573 = 1 (1) reducesto a generating function of 119903-Stirling numbers [8] of thesecond kind and further reduces to a generating functionof the classical Stirling numbers of the second when 119903 =0 Combinatorial interpretation and probability distributioninvolving 119878120573119903(119899119898) are discussed in [9]

The behavior of the numbers 119878120573119903(119899119898) was shown to beasymptotically normal in [4 6]That means that the distribu-tion of these numbers when the parameters 119899 and119898 are largewill follow a bell-shaped distributionThe unimodality of thisdistribution was also discussed in [4] Moreover the boundfor the index in which the maximum value of these numbersoccurs has been established in [10] With these propertiesof the numbers 119878120573119903(119899119898) it is necessary to consider theasymptotic formula for the numbers 119878120573119903(119899119898) to be able tocompute large values of the numbers

Applying Cauchy Integral Formula to (1) the followingintegral representation is obtained

119878120573119903 (119899119898) =119899

2120587119894120573119898 (119898)int119862

119890119903119911(119890120573119911 minus 1)119898

119911119899+1119889119911 (2)

where 119862 is a circle about the originThe primary purpose of the present paper is to investigate

if the analysis in [3] can be extended to the generalized Stir-ling numbers of the second kind The authors are motivatedby the work of Chelluri et al [11] which proved the asymptoticequivalence of the formulas obtained by Temme [1] and thoseobtained byMoser andWyman in [2 3]Moreover it was alsoshown in [11] that the formulas obtained apply not only forintegral values of the parameters 119899 and119898 but for real valuesas well To be able to do a similar investigationwith that in [11]for the generalized Stirling numbers of the first and second

2 Journal of Applied Mathematics

kinds it is necessary to come upwith asymptotic formulas foreach kind generalized Stirling numbers which may have useda method similar to that in [2 3] The necessary asymptoticformulas for the generalized Stirling numbers of the firstkind can be found in [12 13] In this paper an asymptoticformula for the generalized Stirling numbers of the secondkind 119878120573119903(119899119898) with integral values of 119898 and 119899 are obtainedusing a similar analysis as that in [3] The formula is provedto be valid when 119898 gt (14)119899120583 and lim119899rarrinfin(119899 minus 119898) = infinwhere 120583 = 119903120573 Moreover other asymptotic formulas areobtained for 119878120573119903(119899119898) where 119899 and119898 are real numbers underthe conditions 119899 minus 119898 ge 119899120572 and 119899 minus 119898 ge 11989913

2 Preliminary Results

The integral representation in (2) can be written in the form

119878120573119903 (119899119898) =119899

2120587119894120573119898minus119899119898int119862

119890120583119908(119890119908 minus 1)119898

119908119899+1119889119908 (3)

where 120583 = 119903120573 119908 = 120573119911Using the representation119908 = 119877119890119894120579 for the circle 119862 where

minus120587 le 120579 le 120587 and 119877 is a positive real number (3) becomes

119878120573119903 (119899119898) =119899

2120587120573119898minus119899119877119899 (119898)int120587

minus120587

119890120583119877119890119894120579

(119890119877119890119894120579

minus 1)119898

119890119894119899120579119889120579 (4)

Multiplying and dividing the right-hand side of (4) by thecorrection constant

(119890119877minus 1)119898119890120583119877 (5)

it reduces to

119878120573119903 (119899119898) = 119860int120587

minus120587exp ℎ (120579 119877) 119889120579 (6)

where

119860 =119899119890120583119877(119890119877 minus 1)

119898

2120587120573119898minus119899119877119899119898 (7)

and ℎ(120579 119877) is the function

ℎ (120579 119877) = 120583119877119890119894120579+ 119898 log (119890119877119890

119894120579

minus 1)

minus 119898 log (119890119877 minus 1) minus 120583119877 minus 119894119899120579(8)

Observe that ℎ(120579 119877) can be written in the form

ℎ (120579 119877) = 120583119877119890119894120579minus 120583119877 + 119898119892 (120579 119877) (9)

where 119892(120579 119877) is the same function 119892(120579 119877)which appeared in[14] except for the value of 119877

The Maclaurin expansion of ℎ(120579 119877) is

ℎ (120579 119877) = 119894119861120579 + 119898119877119867(119894120579)2+ 119898

infin

sum119896=3

119863119896 (119877) (119894120579)119896 (10)

where

119861 = 119877(120583 +119898

1 minus 119890minus119877) minus 119899 (11)

119867 =1

2[120583

119898+119890119877 (119890119877 minus 119877 minus 1)

(119890119877 minus 1)2

] (12)

119863119896 (119877) =120583119877

119896119898+1

119896Δ119896 log (119890119877 minus 1) (13)

where Δ is the operator 119877(119889119889119877)

Lemma 1 There is a unique 0 lt 119877 lt 119899119898 such that

119877(120583 +119898

1 minus 119890minus119877) minus 119899 = 0 (14)

Proof Equation (14) can be written in the form

119877(1 minus 119890minus119877)minus1=119899

119898minus120583

119898119877 (15)

Let 1198911(119877) = 119877(1 minus 119890minus119877)minus1 and 1198912(119877) = 119899119898 minus (120583119898)119877

Note that 1198911(119877) is a continuous function on (0infin) andlim119877rarrinfin1198911(119877) = infin while lim119877rarr01198911(119877) = 1 On the otherhand 1198912(119877) is a line with 119909 intercept at 119877 = 119899120583 and 119910intercept at 1198912(0) = 119899119898 and is decreasing when 119898 and120583 are positive real numbers Thus 1198911 and 1198912 as functionsof 119877 surely intersect at some point The value of 119877 at theintersection point is the desired solution Moreover it can beseen graphically that 119877 lt 119899120583

Lemma 2 119867 defined in (12) obeys the inequalities

1

4le 119867 le

1

2+120583

2 (16)

Proof This lemma follows from Lemma 22 in [3]

Lemma 3 There exists a constant119872 independent of 119896 and 119877such that

10038161003816100381610038161003816100381610038161003816

119863119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816le 119872 (17)

Proof 119863119896(119877) defined in (13) can be written as

119863119896 (119877) =120583119877

119896119898+ 119862119896 (119877) (18)

where 119862119896(119877) is the same as that defined in (16) in [3]Dividing both sides of the preceding equation by119877 and takingthe absolute value we have

10038161003816100381610038161003816100381610038161003816

119863119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816le1003816100381610038161003816100381610038161003816

120583

119896119898

1003816100381610038161003816100381610038161003816+10038161003816100381610038161003816100381610038161003816

119862119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816 (19)

With 120583 a fixed parameter and 119898 rarr infin as 119899 rarr infin andusing Lemma 32 in [3] the desired result is obtained

Define 120598 = (119898119877)minus38 and consider the integral 119869 definedby

119869 = int120587

120598exp ℎ (120579 119877) 119889120579 (20)

Journal of Applied Mathematics 3

Lemma 4 A constant 119896 gt 0 exists such that

|119869| le 120587 exp (minus119896(119898119877)14) (21)

Proof We claim that

|ℎ (120579 119877)| le100381610038161003816100381610038161003816(119890119877119890119894120579minus 1)

100381610038161003816100381610038161003816sdot100381610038161003816100381610038161003816(119890119877minus 1)minus1100381610038161003816100381610038161003816

119898

(22)

Using the definition of ℎ(120579 119877) given by (9) we have

1003816100381610038161003816exp ℎ (120579 119877)1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816 sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119890119877119890119894120579

minus 1

119890119877 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119898

sdot10038161003816100381610038161003816119890minus11989411989912057910038161003816100381610038161003816

(23)

Because |119890119894119899120579| = 1 it remains to show that | exp[120583119877(119890119894120579 minus1)]| le 1

Note that | exp[120583119877(119890119894120579minus1)]| = 119890120583119877 cos 120579 sdot119890minus120583119877The claim nowfollows from the fact that minus1 le cos 120579 le 1 and 120583119877 ge 0

From (9)

ℎ (120579 119877) = 120583119877 (119890119894120579minus 120583119877) + 119898119892 (120579 119877) (24)

Thus1003816100381610038161003816exp ℎ (120579 119877)

1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816

1003816100381610038161003816exp119898119892 (120579 119877)1003816100381610038161003816 (25)

The result now follows from the previous claim and thefact that 119892(120579 119877) is the same function 119892(120579 119877) that appeared in[3] except for the value of 119877

Observe that when119898119877 rarr infin as 119899 rarr infin Lemma 4 willimply that 119869 rarr 0 as 119899 rarr infin Indeed119898119877 rarr infin as 119899 rarr infin

under some conditions as shown in the following lemma

Lemma 5 119898119877 rarr infin as 119899minus119898 rarr infin provided119898 gt (14)119899120583119899 gt 4

Proof Write (14) in the form

119898119877

1 minus 119890minus119877= 119899 minus 120583119877 (26)

Now the application of the mean value theorem to thefunction 119891(119877) = 1 minus 119890minus119877 over the interval (0 119877) will yield

1 minus 119890minus119877= 119877119890minus120589 (27)

where 120589 is a number within the interval (0 119877) Then119898

119899 minus 120583119877= 119890minus120589 (28)

Consequently

1 minus 119890minus119877= 119877(

119898

119899 minus 120583119877)

1 minus119898119877

119899 minus 120583119877= 119890minus119877le 119890minus120589=

119898

119899 minus 120583119877

1 minus119898119877

119899 minus 120583119877le

119898

119899 minus 120583119877

(29)

The last inequality will yield

119898119877 ge (119899 minus 120583119877) minus 119898 (30)

provided 119899 minus 120583119877 gt 0 The previous inequality can be writtenin the form

119898119877 ge 119899 minus (119898 minus 120583119877) ge 119899 minus (119898 +119899120583

119898) ge 119899 minus (119898 + 4) (31)

when 119898 ge (14)119899120583 The second inequality previous followsif 119877 lt 119899119898 Indeed 119877 lt 119899119898 when 119898 ge (14)119899120583 and119899 gt 4 because 1198911(119899119898) gt 1198912(119899119898) where 1198911 and 1198912 arethe functions in the proof of Lemma 1 Hence 119877 whichis the 119909-coordinate of the point of intersection of the twofunctions must be less than 119899119898 The last inequality showsthat119898119877 rarr infin as 119899 minus 119898 rarr infin under the given restriction of119898

Returning to the condition that 119899 minus 120583119877 gt 0 note that119877 lt 119899119898 Thus

119899 minus 120583119877 gt 119899 minus 120583119899

119898gt 119899 minus 4 (32)

under the restriction that 119898 ge (14)119899120583 So that 119899 minus 120583119877 gt 0whenever 119899 gt 4

For example when 119899 = 100 and 120583 = 47 119898 in Lemma 5must satisfy119898 ge (14)100(47) = 142857 Thus119898 ge 15

3 Asymptotic Formula withIntegral Parameters

In the discussion that follows 119877 denotes the unique positivesolution to (14) It will be seen later that our final asymptoticformula is expressed in terms of powers of 1119898119877 Anticipat-ing this result and in view of Lemma 4 we write

119878120573119903 (119898 119899) sim 119860int120598

minus120598exp ℎ (120579 119877) 119889120579 (33)

Substituting (10) for ℎ(120579 119877) and noting that119861(119877) = 0 (33)becomes

119878120573119903 (119899119898) sim 119860int120598

minus120598exp[minus1198981198771198671205792 + 119898

infin

sum119896=3

119863119896 (119877) (119894120579)119896]119889120579

(34)

The change of variable 120601 = (119898119877119867)12120579 will yield 119889120601 =(119898119877119867)

12119889120579

120579 =120601

(119898119877119867)12 119889120579 =

119889120601

(119898119877119867)12 (35)

Now (34) becomes

119878120573119903 (119899119898) sim119860

(119898119877119867)12int120572

minus120572119890minus1206012 exp [119891 (119911 119877 120601)] 119889120601 (36)

where 120572 = (119898119877119867)12120598 119911 = (119898119877119867)minus12

119891 (119911 119877 120601) =

infin

sum119896=1

119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2(119898119877)minus1198962 (37)

4 Journal of Applied Mathematics

The equation in (37) can be written in the from

119891 (119911 119877 120601) =

infin

sum119896=1

119886119896119911119896 (38)

where

119886119896 = 119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2 (39)

Note here that 119911 = (119898119877)minus12 is within the radius of

convergence of the series in (38) On the other hand theMaclaurin series of exp119891(119911 119877 120601) is

119890119891(119911119877120601)

=

infin

sum119896=0

119887119896 (119877 120601) 119911119896 (40)

where 1198872119896 is a polynomial in 120601 containing only even powers of120601 and 1198872119896+1 is a polynomial in 120601 containing only odd powersof 120601 Now (36) can be written in the form

119878120573119903 (119899119898) sim 119876int120572

minus120572119890minus1206012(

infin

sum119896=0

119887119896 (119877 120601) 119911119896)119889120601 (41)

where 119876 = 119860radic119898119877119867 and 119911 = (119898119877)minus12We write (41) as

119878120573119903 (119899119898) sim 119876[

119904minus1

sum119896=0

119911119896(int120572

minus120572119890minus1206012119887119896119889120601) + 119880119904] (42)

where

119880119904 =

infin

sum119896=119904

(int120572

minus120572119890minus1206012119887119896119889120601) 119911

119896 (43)

In view of Lemma 2119867 ge 14 hence

120572 = (119898119877119867)12120598 = (119898119877119867)

12(119898119877)minus38

ge(119898119877)18

2 (44)

Since 119887119896 is a polynomial in 120601 we can replace 120572 with infinin (41) Following the discussion in [3] it can be shown that119880119904 = 119874(119911

119904) hence we have the asymptotic formula

119878120573119903 (119899119898) sim 119876[

infin

sum119896=0

(intinfin

minusinfin119890minus12060121198872119896119889120601) (119898119877)

minus119896] (45)

Note that

intinfin

minusinfin119890minus12060121198872119896+1119889120601 = 0 (46)

This is why no odd subscript of 119887119896 appears in (45)An approximation is obtained by taking the first two

terms of the sum in (45) Thus

119878120573119903 (119899119898) asymp 119876[intinfin

minusinfin119890minus12060121198870119889120601 + (119898119877)

minus1intinfin

minusinfin119890minus12060121198872119889120601]

(47)

It can be computed that

1198870 = 1

intinfin

minusinfin119890minus1206012119889120601 = radic120587

1198872 = 1198862 (119877 120601) +1

2[1198861 (119877 120601)]

2

1198862 = 1198634 (119877)(119894120601)4

1198771198672

1198634 (119877) =120583119877

24119898+ 1198624 (119877)

(48)

where

1198624 (119877) =1

24[119877 + (119877 minus 7119877

2+ 61198773minus 1198774) (119890119877minus 1)minus1]

+1

24[(18119877

3minus 71198772minus 71198774) (119890119877minus 1)minus2

+ (121198773minus 12119877

4) (119890119877minus 1)minus3]

minus1

24[61198774(119890119877minus 1)minus4]

(49)

while

1198861 (119877 120601) = 1198633(119894120601)3

11987711986732= (

120583119877

3119898+ 1198623)

(119894120601)3

11987711986732 (50)

where

1198623 =1

6[119877 + (119877 minus 3119877

2+ 1198773) (119890119877minus 1)minus1

+ (31198773minus 31198772) (119890119877minus 1)minus2]

+1

6[21198773(119890119877minus 1)minus3]

(51)

Thus 1198872 can be written as follows

1198872 = [120583119877

24119898+ 1198624]

1206014

1198771198672minus1

2(120583119877

6119898+ 1198623)

2 1206016

11987721198673 (52)

Let

119868 = intinfin

minusinfin119890minus12060121198872119889120601 (53)

Journal of Applied Mathematics 5

Then

119868 = intinfin

minusinfin119890minus1206012 (12058311987724119898 + 1198624)

11987711986721206014119889120601

minus1

2intinfin

minusinfin119890minus1206012 (1205831198776119898 + 1198623)

2

119877211986731206016119889120601

=12058311987724119898 + 1198624

1198771198672intinfin

minusinfin119890minus12060121206014119889120601

minus(1205831198776119898 + 1198623)

2

211987721198673intinfin

minusinfin119890minus12060121206016119889120601

=12058311987724119898 + 1198624

1198771198672sdot3radic120587

4minus(1205831198776119898 + 1198623)

2

11987721198673sdot15radic120587

16

(54)

Finally we obtain the approximation formula

119878120573119903 (119899119898) asymp119899

119898

119890120583119877(119890119877 minus 1)119898

2120573119898minus119899119877119899radic120587119898119877119867[1 +

119868

119898119877radic120587] (55)

In view of Lemmas 4 and 5 and (33) the previousasymptotic approximation is valid for 119898 ge (14)119899120583 Theformula obtained in the previous discussion is formally statedin the following theorem

Theorem 6 The formula

119878120573119903 (119899119898) sim 119876 [radic120587 +119868

119898119877] (56)

behaves as an asymptotic approximation for the generalizedStirling numbers of the second kind 119878120573119903(119899119898) with positiveintegral parameters for 119898 gt (14)119899120583 119899 gt 4 and such that119899 minus 119898 rarr infin as 119899 rarr infin

Table 1 displays the exact values and approximate valuesfor 119899 = 100 119903 = 4 120573 = 7 The exact values are obtainedusing recurrence formula while the approximate values areobtained using Theorem 6 In view of Lemma 5 the formulais valid for119898 ge 15

Values on the table affirm that the asymptotic formula inTheorem 6 gives a good approximation for 11987874(100119898)when119898 ge 15

4 Asymptotic Formula with Real Parameters

Recently the (119903 120573)-Stirling numbers for complex argumentswere defined in [15] parallel to the definition of Flajolet andProdinger as

119878120573119903 (119909 119910) =119909

1199101205731199102120587119894intH

119890119903119911(119890120573119911minus 1)119910 119889119911

119911119909+1 (57)

where 119909 = Γ(119909 + 1) and H is a Hankel contour thatstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeI119911 gt 0

Table 1

Exact value Approximate value Relative error11987874(100 5) 5685 times 10152 6335 times 10152 011428

11987874(100 10) 7728 times 10171

8169 times 10171

005713

11987874(100 15) 8411 times 10178 8731 times 10178 003804

11987874(100 30) 5604 times 10174 5706 times 10174 001824

11987874(100 60) 7399 times 10122

7446 times 10122

000641

11987874(100 80) 1275 times 1070

1279 times 1070

000263

11987874(100 90) 2208 times 1038 2211 times 1038 000123

Note that by change of variable say 119908 = 120573119911 we can express119878120573119903(119909 119910) as follows

119878120573119903 (119909 119910) = 120573119909minus119910

119909 +119903

120573

119910 +119903

120573

119903120573

(58)

where

119909 + 119903

119910 + 119903119903

=119909

1199102120587119894intH

119890119903119911(119890119911minus 1)119910 119889119911

119911119909+1 (59)

The numbers 119909+119903119910+119903 119903 are certain generalization of 119903-Stirling numbers of the second kind [8] in which the param-eters involved are complex numbers These numbers satisfythe following properties

Theorem 7 For nonnegative real number 119903 one has

119909 + 119903

119910 + 119903119903

= 119909 + 119903 minus 1

119910 + 119903 minus 1119903

+ (119910 + 119903) 119909 + 119903 minus 1

119910 + 119903119903

(60)

Theorem 8 For nonnegative real numbers 119903 and 120573 and 119896 isin None has

119909 + 119903

119896 + 119903119903

=1

119896

119896

sum119895=0

(minus1)119896minus119895(119896

119895) (119895 + 119903)

119909 (61)

Furthermore when 119896 = 119910 a complex number

119909 + 119903

119910 + 119903119903

=1

119910

infin

sum119895=0

(minus1)119910minus119895(119910

119895) (119895 + 119903)

119909 (62)

The Bernoulli polynomial can be expressed using theexplicit formula in [16] and the first formula in Theorem 8with 119908 = 119903 and 119909 = 119899 as

119861119899 (119903) =

infin

sum119896=0

(minus1)119896119896

119896 + 1119899 + 119903

119896 + 119903119903

(63)

Moreover it is known that for 119899 = 1 2 the Hurwitzzeta function 120577(1 minus 119899 119909) = minus119861119899(119909)119899 Note that as 119910 rarr 0(119910 minus 1) sim 1119910 By Cauchyrsquos integral formula

119899 + 119903

119910 + 119903 minus 1119903

sim 119910119899 [119911119899+1]119890119903119911

119890119911 minus 1= 119910119861119899+1 (119903)

119899 + 1 (64)

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

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DifferentialEquations

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2 Journal of Applied Mathematics

kinds it is necessary to come upwith asymptotic formulas foreach kind generalized Stirling numbers which may have useda method similar to that in [2 3] The necessary asymptoticformulas for the generalized Stirling numbers of the firstkind can be found in [12 13] In this paper an asymptoticformula for the generalized Stirling numbers of the secondkind 119878120573119903(119899119898) with integral values of 119898 and 119899 are obtainedusing a similar analysis as that in [3] The formula is provedto be valid when 119898 gt (14)119899120583 and lim119899rarrinfin(119899 minus 119898) = infinwhere 120583 = 119903120573 Moreover other asymptotic formulas areobtained for 119878120573119903(119899119898) where 119899 and119898 are real numbers underthe conditions 119899 minus 119898 ge 119899120572 and 119899 minus 119898 ge 11989913

2 Preliminary Results

The integral representation in (2) can be written in the form

119878120573119903 (119899119898) =119899

2120587119894120573119898minus119899119898int119862

119890120583119908(119890119908 minus 1)119898

119908119899+1119889119908 (3)

where 120583 = 119903120573 119908 = 120573119911Using the representation119908 = 119877119890119894120579 for the circle 119862 where

minus120587 le 120579 le 120587 and 119877 is a positive real number (3) becomes

119878120573119903 (119899119898) =119899

2120587120573119898minus119899119877119899 (119898)int120587

minus120587

119890120583119877119890119894120579

(119890119877119890119894120579

minus 1)119898

119890119894119899120579119889120579 (4)

Multiplying and dividing the right-hand side of (4) by thecorrection constant

(119890119877minus 1)119898119890120583119877 (5)

it reduces to

119878120573119903 (119899119898) = 119860int120587

minus120587exp ℎ (120579 119877) 119889120579 (6)

where

119860 =119899119890120583119877(119890119877 minus 1)

119898

2120587120573119898minus119899119877119899119898 (7)

and ℎ(120579 119877) is the function

ℎ (120579 119877) = 120583119877119890119894120579+ 119898 log (119890119877119890

119894120579

minus 1)

minus 119898 log (119890119877 minus 1) minus 120583119877 minus 119894119899120579(8)

Observe that ℎ(120579 119877) can be written in the form

ℎ (120579 119877) = 120583119877119890119894120579minus 120583119877 + 119898119892 (120579 119877) (9)

where 119892(120579 119877) is the same function 119892(120579 119877)which appeared in[14] except for the value of 119877

The Maclaurin expansion of ℎ(120579 119877) is

ℎ (120579 119877) = 119894119861120579 + 119898119877119867(119894120579)2+ 119898

infin

sum119896=3

119863119896 (119877) (119894120579)119896 (10)

where

119861 = 119877(120583 +119898

1 minus 119890minus119877) minus 119899 (11)

119867 =1

2[120583

119898+119890119877 (119890119877 minus 119877 minus 1)

(119890119877 minus 1)2

] (12)

119863119896 (119877) =120583119877

119896119898+1

119896Δ119896 log (119890119877 minus 1) (13)

where Δ is the operator 119877(119889119889119877)

Lemma 1 There is a unique 0 lt 119877 lt 119899119898 such that

119877(120583 +119898

1 minus 119890minus119877) minus 119899 = 0 (14)

Proof Equation (14) can be written in the form

119877(1 minus 119890minus119877)minus1=119899

119898minus120583

119898119877 (15)

Let 1198911(119877) = 119877(1 minus 119890minus119877)minus1 and 1198912(119877) = 119899119898 minus (120583119898)119877

Note that 1198911(119877) is a continuous function on (0infin) andlim119877rarrinfin1198911(119877) = infin while lim119877rarr01198911(119877) = 1 On the otherhand 1198912(119877) is a line with 119909 intercept at 119877 = 119899120583 and 119910intercept at 1198912(0) = 119899119898 and is decreasing when 119898 and120583 are positive real numbers Thus 1198911 and 1198912 as functionsof 119877 surely intersect at some point The value of 119877 at theintersection point is the desired solution Moreover it can beseen graphically that 119877 lt 119899120583

Lemma 2 119867 defined in (12) obeys the inequalities

1

4le 119867 le

1

2+120583

2 (16)

Proof This lemma follows from Lemma 22 in [3]

Lemma 3 There exists a constant119872 independent of 119896 and 119877such that

10038161003816100381610038161003816100381610038161003816

119863119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816le 119872 (17)

Proof 119863119896(119877) defined in (13) can be written as

119863119896 (119877) =120583119877

119896119898+ 119862119896 (119877) (18)

where 119862119896(119877) is the same as that defined in (16) in [3]Dividing both sides of the preceding equation by119877 and takingthe absolute value we have

10038161003816100381610038161003816100381610038161003816

119863119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816le1003816100381610038161003816100381610038161003816

120583

119896119898

1003816100381610038161003816100381610038161003816+10038161003816100381610038161003816100381610038161003816

119862119896 (119877)

119877

10038161003816100381610038161003816100381610038161003816 (19)

With 120583 a fixed parameter and 119898 rarr infin as 119899 rarr infin andusing Lemma 32 in [3] the desired result is obtained

Define 120598 = (119898119877)minus38 and consider the integral 119869 definedby

119869 = int120587

120598exp ℎ (120579 119877) 119889120579 (20)

Journal of Applied Mathematics 3

Lemma 4 A constant 119896 gt 0 exists such that

|119869| le 120587 exp (minus119896(119898119877)14) (21)

Proof We claim that

|ℎ (120579 119877)| le100381610038161003816100381610038161003816(119890119877119890119894120579minus 1)

100381610038161003816100381610038161003816sdot100381610038161003816100381610038161003816(119890119877minus 1)minus1100381610038161003816100381610038161003816

119898

(22)

Using the definition of ℎ(120579 119877) given by (9) we have

1003816100381610038161003816exp ℎ (120579 119877)1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816 sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119890119877119890119894120579

minus 1

119890119877 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119898

sdot10038161003816100381610038161003816119890minus11989411989912057910038161003816100381610038161003816

(23)

Because |119890119894119899120579| = 1 it remains to show that | exp[120583119877(119890119894120579 minus1)]| le 1

Note that | exp[120583119877(119890119894120579minus1)]| = 119890120583119877 cos 120579 sdot119890minus120583119877The claim nowfollows from the fact that minus1 le cos 120579 le 1 and 120583119877 ge 0

From (9)

ℎ (120579 119877) = 120583119877 (119890119894120579minus 120583119877) + 119898119892 (120579 119877) (24)

Thus1003816100381610038161003816exp ℎ (120579 119877)

1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816

1003816100381610038161003816exp119898119892 (120579 119877)1003816100381610038161003816 (25)

The result now follows from the previous claim and thefact that 119892(120579 119877) is the same function 119892(120579 119877) that appeared in[3] except for the value of 119877

Observe that when119898119877 rarr infin as 119899 rarr infin Lemma 4 willimply that 119869 rarr 0 as 119899 rarr infin Indeed119898119877 rarr infin as 119899 rarr infin

under some conditions as shown in the following lemma

Lemma 5 119898119877 rarr infin as 119899minus119898 rarr infin provided119898 gt (14)119899120583119899 gt 4

Proof Write (14) in the form

119898119877

1 minus 119890minus119877= 119899 minus 120583119877 (26)

Now the application of the mean value theorem to thefunction 119891(119877) = 1 minus 119890minus119877 over the interval (0 119877) will yield

1 minus 119890minus119877= 119877119890minus120589 (27)

where 120589 is a number within the interval (0 119877) Then119898

119899 minus 120583119877= 119890minus120589 (28)

Consequently

1 minus 119890minus119877= 119877(

119898

119899 minus 120583119877)

1 minus119898119877

119899 minus 120583119877= 119890minus119877le 119890minus120589=

119898

119899 minus 120583119877

1 minus119898119877

119899 minus 120583119877le

119898

119899 minus 120583119877

(29)

The last inequality will yield

119898119877 ge (119899 minus 120583119877) minus 119898 (30)

provided 119899 minus 120583119877 gt 0 The previous inequality can be writtenin the form

119898119877 ge 119899 minus (119898 minus 120583119877) ge 119899 minus (119898 +119899120583

119898) ge 119899 minus (119898 + 4) (31)

when 119898 ge (14)119899120583 The second inequality previous followsif 119877 lt 119899119898 Indeed 119877 lt 119899119898 when 119898 ge (14)119899120583 and119899 gt 4 because 1198911(119899119898) gt 1198912(119899119898) where 1198911 and 1198912 arethe functions in the proof of Lemma 1 Hence 119877 whichis the 119909-coordinate of the point of intersection of the twofunctions must be less than 119899119898 The last inequality showsthat119898119877 rarr infin as 119899 minus 119898 rarr infin under the given restriction of119898

Returning to the condition that 119899 minus 120583119877 gt 0 note that119877 lt 119899119898 Thus

119899 minus 120583119877 gt 119899 minus 120583119899

119898gt 119899 minus 4 (32)

under the restriction that 119898 ge (14)119899120583 So that 119899 minus 120583119877 gt 0whenever 119899 gt 4

For example when 119899 = 100 and 120583 = 47 119898 in Lemma 5must satisfy119898 ge (14)100(47) = 142857 Thus119898 ge 15

3 Asymptotic Formula withIntegral Parameters

In the discussion that follows 119877 denotes the unique positivesolution to (14) It will be seen later that our final asymptoticformula is expressed in terms of powers of 1119898119877 Anticipat-ing this result and in view of Lemma 4 we write

119878120573119903 (119898 119899) sim 119860int120598

minus120598exp ℎ (120579 119877) 119889120579 (33)

Substituting (10) for ℎ(120579 119877) and noting that119861(119877) = 0 (33)becomes

119878120573119903 (119899119898) sim 119860int120598

minus120598exp[minus1198981198771198671205792 + 119898

infin

sum119896=3

119863119896 (119877) (119894120579)119896]119889120579

(34)

The change of variable 120601 = (119898119877119867)12120579 will yield 119889120601 =(119898119877119867)

12119889120579

120579 =120601

(119898119877119867)12 119889120579 =

119889120601

(119898119877119867)12 (35)

Now (34) becomes

119878120573119903 (119899119898) sim119860

(119898119877119867)12int120572

minus120572119890minus1206012 exp [119891 (119911 119877 120601)] 119889120601 (36)

where 120572 = (119898119877119867)12120598 119911 = (119898119877119867)minus12

119891 (119911 119877 120601) =

infin

sum119896=1

119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2(119898119877)minus1198962 (37)

4 Journal of Applied Mathematics

The equation in (37) can be written in the from

119891 (119911 119877 120601) =

infin

sum119896=1

119886119896119911119896 (38)

where

119886119896 = 119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2 (39)

Note here that 119911 = (119898119877)minus12 is within the radius of

convergence of the series in (38) On the other hand theMaclaurin series of exp119891(119911 119877 120601) is

119890119891(119911119877120601)

=

infin

sum119896=0

119887119896 (119877 120601) 119911119896 (40)

where 1198872119896 is a polynomial in 120601 containing only even powers of120601 and 1198872119896+1 is a polynomial in 120601 containing only odd powersof 120601 Now (36) can be written in the form

119878120573119903 (119899119898) sim 119876int120572

minus120572119890minus1206012(

infin

sum119896=0

119887119896 (119877 120601) 119911119896)119889120601 (41)

where 119876 = 119860radic119898119877119867 and 119911 = (119898119877)minus12We write (41) as

119878120573119903 (119899119898) sim 119876[

119904minus1

sum119896=0

119911119896(int120572

minus120572119890minus1206012119887119896119889120601) + 119880119904] (42)

where

119880119904 =

infin

sum119896=119904

(int120572

minus120572119890minus1206012119887119896119889120601) 119911

119896 (43)

In view of Lemma 2119867 ge 14 hence

120572 = (119898119877119867)12120598 = (119898119877119867)

12(119898119877)minus38

ge(119898119877)18

2 (44)

Since 119887119896 is a polynomial in 120601 we can replace 120572 with infinin (41) Following the discussion in [3] it can be shown that119880119904 = 119874(119911

119904) hence we have the asymptotic formula

119878120573119903 (119899119898) sim 119876[

infin

sum119896=0

(intinfin

minusinfin119890minus12060121198872119896119889120601) (119898119877)

minus119896] (45)

Note that

intinfin

minusinfin119890minus12060121198872119896+1119889120601 = 0 (46)

This is why no odd subscript of 119887119896 appears in (45)An approximation is obtained by taking the first two

terms of the sum in (45) Thus

119878120573119903 (119899119898) asymp 119876[intinfin

minusinfin119890minus12060121198870119889120601 + (119898119877)

minus1intinfin

minusinfin119890minus12060121198872119889120601]

(47)

It can be computed that

1198870 = 1

intinfin

minusinfin119890minus1206012119889120601 = radic120587

1198872 = 1198862 (119877 120601) +1

2[1198861 (119877 120601)]

2

1198862 = 1198634 (119877)(119894120601)4

1198771198672

1198634 (119877) =120583119877

24119898+ 1198624 (119877)

(48)

where

1198624 (119877) =1

24[119877 + (119877 minus 7119877

2+ 61198773minus 1198774) (119890119877minus 1)minus1]

+1

24[(18119877

3minus 71198772minus 71198774) (119890119877minus 1)minus2

+ (121198773minus 12119877

4) (119890119877minus 1)minus3]

minus1

24[61198774(119890119877minus 1)minus4]

(49)

while

1198861 (119877 120601) = 1198633(119894120601)3

11987711986732= (

120583119877

3119898+ 1198623)

(119894120601)3

11987711986732 (50)

where

1198623 =1

6[119877 + (119877 minus 3119877

2+ 1198773) (119890119877minus 1)minus1

+ (31198773minus 31198772) (119890119877minus 1)minus2]

+1

6[21198773(119890119877minus 1)minus3]

(51)

Thus 1198872 can be written as follows

1198872 = [120583119877

24119898+ 1198624]

1206014

1198771198672minus1

2(120583119877

6119898+ 1198623)

2 1206016

11987721198673 (52)

Let

119868 = intinfin

minusinfin119890minus12060121198872119889120601 (53)

Journal of Applied Mathematics 5

Then

119868 = intinfin

minusinfin119890minus1206012 (12058311987724119898 + 1198624)

11987711986721206014119889120601

minus1

2intinfin

minusinfin119890minus1206012 (1205831198776119898 + 1198623)

2

119877211986731206016119889120601

=12058311987724119898 + 1198624

1198771198672intinfin

minusinfin119890minus12060121206014119889120601

minus(1205831198776119898 + 1198623)

2

211987721198673intinfin

minusinfin119890minus12060121206016119889120601

=12058311987724119898 + 1198624

1198771198672sdot3radic120587

4minus(1205831198776119898 + 1198623)

2

11987721198673sdot15radic120587

16

(54)

Finally we obtain the approximation formula

119878120573119903 (119899119898) asymp119899

119898

119890120583119877(119890119877 minus 1)119898

2120573119898minus119899119877119899radic120587119898119877119867[1 +

119868

119898119877radic120587] (55)

In view of Lemmas 4 and 5 and (33) the previousasymptotic approximation is valid for 119898 ge (14)119899120583 Theformula obtained in the previous discussion is formally statedin the following theorem

Theorem 6 The formula

119878120573119903 (119899119898) sim 119876 [radic120587 +119868

119898119877] (56)

behaves as an asymptotic approximation for the generalizedStirling numbers of the second kind 119878120573119903(119899119898) with positiveintegral parameters for 119898 gt (14)119899120583 119899 gt 4 and such that119899 minus 119898 rarr infin as 119899 rarr infin

Table 1 displays the exact values and approximate valuesfor 119899 = 100 119903 = 4 120573 = 7 The exact values are obtainedusing recurrence formula while the approximate values areobtained using Theorem 6 In view of Lemma 5 the formulais valid for119898 ge 15

Values on the table affirm that the asymptotic formula inTheorem 6 gives a good approximation for 11987874(100119898)when119898 ge 15

4 Asymptotic Formula with Real Parameters

Recently the (119903 120573)-Stirling numbers for complex argumentswere defined in [15] parallel to the definition of Flajolet andProdinger as

119878120573119903 (119909 119910) =119909

1199101205731199102120587119894intH

119890119903119911(119890120573119911minus 1)119910 119889119911

119911119909+1 (57)

where 119909 = Γ(119909 + 1) and H is a Hankel contour thatstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeI119911 gt 0

Table 1

Exact value Approximate value Relative error11987874(100 5) 5685 times 10152 6335 times 10152 011428

11987874(100 10) 7728 times 10171

8169 times 10171

005713

11987874(100 15) 8411 times 10178 8731 times 10178 003804

11987874(100 30) 5604 times 10174 5706 times 10174 001824

11987874(100 60) 7399 times 10122

7446 times 10122

000641

11987874(100 80) 1275 times 1070

1279 times 1070

000263

11987874(100 90) 2208 times 1038 2211 times 1038 000123

Note that by change of variable say 119908 = 120573119911 we can express119878120573119903(119909 119910) as follows

119878120573119903 (119909 119910) = 120573119909minus119910

119909 +119903

120573

119910 +119903

120573

119903120573

(58)

where

119909 + 119903

119910 + 119903119903

=119909

1199102120587119894intH

119890119903119911(119890119911minus 1)119910 119889119911

119911119909+1 (59)

The numbers 119909+119903119910+119903 119903 are certain generalization of 119903-Stirling numbers of the second kind [8] in which the param-eters involved are complex numbers These numbers satisfythe following properties

Theorem 7 For nonnegative real number 119903 one has

119909 + 119903

119910 + 119903119903

= 119909 + 119903 minus 1

119910 + 119903 minus 1119903

+ (119910 + 119903) 119909 + 119903 minus 1

119910 + 119903119903

(60)

Theorem 8 For nonnegative real numbers 119903 and 120573 and 119896 isin None has

119909 + 119903

119896 + 119903119903

=1

119896

119896

sum119895=0

(minus1)119896minus119895(119896

119895) (119895 + 119903)

119909 (61)

Furthermore when 119896 = 119910 a complex number

119909 + 119903

119910 + 119903119903

=1

119910

infin

sum119895=0

(minus1)119910minus119895(119910

119895) (119895 + 119903)

119909 (62)

The Bernoulli polynomial can be expressed using theexplicit formula in [16] and the first formula in Theorem 8with 119908 = 119903 and 119909 = 119899 as

119861119899 (119903) =

infin

sum119896=0

(minus1)119896119896

119896 + 1119899 + 119903

119896 + 119903119903

(63)

Moreover it is known that for 119899 = 1 2 the Hurwitzzeta function 120577(1 minus 119899 119909) = minus119861119899(119909)119899 Note that as 119910 rarr 0(119910 minus 1) sim 1119910 By Cauchyrsquos integral formula

119899 + 119903

119910 + 119903 minus 1119903

sim 119910119899 [119911119899+1]119890119903119911

119890119911 minus 1= 119910119861119899+1 (119903)

119899 + 1 (64)

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

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DifferentialEquations

International Journal of

Volume 2013

Journal of Applied Mathematics 3

Lemma 4 A constant 119896 gt 0 exists such that

|119869| le 120587 exp (minus119896(119898119877)14) (21)

Proof We claim that

|ℎ (120579 119877)| le100381610038161003816100381610038161003816(119890119877119890119894120579minus 1)

100381610038161003816100381610038161003816sdot100381610038161003816100381610038161003816(119890119877minus 1)minus1100381610038161003816100381610038161003816

119898

(22)

Using the definition of ℎ(120579 119877) given by (9) we have

1003816100381610038161003816exp ℎ (120579 119877)1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816 sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119890119877119890119894120579

minus 1

119890119877 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119898

sdot10038161003816100381610038161003816119890minus11989411989912057910038161003816100381610038161003816

(23)

Because |119890119894119899120579| = 1 it remains to show that | exp[120583119877(119890119894120579 minus1)]| le 1

Note that | exp[120583119877(119890119894120579minus1)]| = 119890120583119877 cos 120579 sdot119890minus120583119877The claim nowfollows from the fact that minus1 le cos 120579 le 1 and 120583119877 ge 0

From (9)

ℎ (120579 119877) = 120583119877 (119890119894120579minus 120583119877) + 119898119892 (120579 119877) (24)

Thus1003816100381610038161003816exp ℎ (120579 119877)

1003816100381610038161003816 =10038161003816100381610038161003816exp [120583119877 (119890119894120579 minus 1)]10038161003816100381610038161003816

1003816100381610038161003816exp119898119892 (120579 119877)1003816100381610038161003816 (25)

The result now follows from the previous claim and thefact that 119892(120579 119877) is the same function 119892(120579 119877) that appeared in[3] except for the value of 119877

Observe that when119898119877 rarr infin as 119899 rarr infin Lemma 4 willimply that 119869 rarr 0 as 119899 rarr infin Indeed119898119877 rarr infin as 119899 rarr infin

under some conditions as shown in the following lemma

Lemma 5 119898119877 rarr infin as 119899minus119898 rarr infin provided119898 gt (14)119899120583119899 gt 4

Proof Write (14) in the form

119898119877

1 minus 119890minus119877= 119899 minus 120583119877 (26)

Now the application of the mean value theorem to thefunction 119891(119877) = 1 minus 119890minus119877 over the interval (0 119877) will yield

1 minus 119890minus119877= 119877119890minus120589 (27)

where 120589 is a number within the interval (0 119877) Then119898

119899 minus 120583119877= 119890minus120589 (28)

Consequently

1 minus 119890minus119877= 119877(

119898

119899 minus 120583119877)

1 minus119898119877

119899 minus 120583119877= 119890minus119877le 119890minus120589=

119898

119899 minus 120583119877

1 minus119898119877

119899 minus 120583119877le

119898

119899 minus 120583119877

(29)

The last inequality will yield

119898119877 ge (119899 minus 120583119877) minus 119898 (30)

provided 119899 minus 120583119877 gt 0 The previous inequality can be writtenin the form

119898119877 ge 119899 minus (119898 minus 120583119877) ge 119899 minus (119898 +119899120583

119898) ge 119899 minus (119898 + 4) (31)

when 119898 ge (14)119899120583 The second inequality previous followsif 119877 lt 119899119898 Indeed 119877 lt 119899119898 when 119898 ge (14)119899120583 and119899 gt 4 because 1198911(119899119898) gt 1198912(119899119898) where 1198911 and 1198912 arethe functions in the proof of Lemma 1 Hence 119877 whichis the 119909-coordinate of the point of intersection of the twofunctions must be less than 119899119898 The last inequality showsthat119898119877 rarr infin as 119899 minus 119898 rarr infin under the given restriction of119898

Returning to the condition that 119899 minus 120583119877 gt 0 note that119877 lt 119899119898 Thus

119899 minus 120583119877 gt 119899 minus 120583119899

119898gt 119899 minus 4 (32)

under the restriction that 119898 ge (14)119899120583 So that 119899 minus 120583119877 gt 0whenever 119899 gt 4

For example when 119899 = 100 and 120583 = 47 119898 in Lemma 5must satisfy119898 ge (14)100(47) = 142857 Thus119898 ge 15

3 Asymptotic Formula withIntegral Parameters

In the discussion that follows 119877 denotes the unique positivesolution to (14) It will be seen later that our final asymptoticformula is expressed in terms of powers of 1119898119877 Anticipat-ing this result and in view of Lemma 4 we write

119878120573119903 (119898 119899) sim 119860int120598

minus120598exp ℎ (120579 119877) 119889120579 (33)

Substituting (10) for ℎ(120579 119877) and noting that119861(119877) = 0 (33)becomes

119878120573119903 (119899119898) sim 119860int120598

minus120598exp[minus1198981198771198671205792 + 119898

infin

sum119896=3

119863119896 (119877) (119894120579)119896]119889120579

(34)

The change of variable 120601 = (119898119877119867)12120579 will yield 119889120601 =(119898119877119867)

12119889120579

120579 =120601

(119898119877119867)12 119889120579 =

119889120601

(119898119877119867)12 (35)

Now (34) becomes

119878120573119903 (119899119898) sim119860

(119898119877119867)12int120572

minus120572119890minus1206012 exp [119891 (119911 119877 120601)] 119889120601 (36)

where 120572 = (119898119877119867)12120598 119911 = (119898119877119867)minus12

119891 (119911 119877 120601) =

infin

sum119896=1

119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2(119898119877)minus1198962 (37)

4 Journal of Applied Mathematics

The equation in (37) can be written in the from

119891 (119911 119877 120601) =

infin

sum119896=1

119886119896119911119896 (38)

where

119886119896 = 119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2 (39)

Note here that 119911 = (119898119877)minus12 is within the radius of

convergence of the series in (38) On the other hand theMaclaurin series of exp119891(119911 119877 120601) is

119890119891(119911119877120601)

=

infin

sum119896=0

119887119896 (119877 120601) 119911119896 (40)

where 1198872119896 is a polynomial in 120601 containing only even powers of120601 and 1198872119896+1 is a polynomial in 120601 containing only odd powersof 120601 Now (36) can be written in the form

119878120573119903 (119899119898) sim 119876int120572

minus120572119890minus1206012(

infin

sum119896=0

119887119896 (119877 120601) 119911119896)119889120601 (41)

where 119876 = 119860radic119898119877119867 and 119911 = (119898119877)minus12We write (41) as

119878120573119903 (119899119898) sim 119876[

119904minus1

sum119896=0

119911119896(int120572

minus120572119890minus1206012119887119896119889120601) + 119880119904] (42)

where

119880119904 =

infin

sum119896=119904

(int120572

minus120572119890minus1206012119887119896119889120601) 119911

119896 (43)

In view of Lemma 2119867 ge 14 hence

120572 = (119898119877119867)12120598 = (119898119877119867)

12(119898119877)minus38

ge(119898119877)18

2 (44)

Since 119887119896 is a polynomial in 120601 we can replace 120572 with infinin (41) Following the discussion in [3] it can be shown that119880119904 = 119874(119911

119904) hence we have the asymptotic formula

119878120573119903 (119899119898) sim 119876[

infin

sum119896=0

(intinfin

minusinfin119890minus12060121198872119896119889120601) (119898119877)

minus119896] (45)

Note that

intinfin

minusinfin119890minus12060121198872119896+1119889120601 = 0 (46)

This is why no odd subscript of 119887119896 appears in (45)An approximation is obtained by taking the first two

terms of the sum in (45) Thus

119878120573119903 (119899119898) asymp 119876[intinfin

minusinfin119890minus12060121198870119889120601 + (119898119877)

minus1intinfin

minusinfin119890minus12060121198872119889120601]

(47)

It can be computed that

1198870 = 1

intinfin

minusinfin119890minus1206012119889120601 = radic120587

1198872 = 1198862 (119877 120601) +1

2[1198861 (119877 120601)]

2

1198862 = 1198634 (119877)(119894120601)4

1198771198672

1198634 (119877) =120583119877

24119898+ 1198624 (119877)

(48)

where

1198624 (119877) =1

24[119877 + (119877 minus 7119877

2+ 61198773minus 1198774) (119890119877minus 1)minus1]

+1

24[(18119877

3minus 71198772minus 71198774) (119890119877minus 1)minus2

+ (121198773minus 12119877

4) (119890119877minus 1)minus3]

minus1

24[61198774(119890119877minus 1)minus4]

(49)

while

1198861 (119877 120601) = 1198633(119894120601)3

11987711986732= (

120583119877

3119898+ 1198623)

(119894120601)3

11987711986732 (50)

where

1198623 =1

6[119877 + (119877 minus 3119877

2+ 1198773) (119890119877minus 1)minus1

+ (31198773minus 31198772) (119890119877minus 1)minus2]

+1

6[21198773(119890119877minus 1)minus3]

(51)

Thus 1198872 can be written as follows

1198872 = [120583119877

24119898+ 1198624]

1206014

1198771198672minus1

2(120583119877

6119898+ 1198623)

2 1206016

11987721198673 (52)

Let

119868 = intinfin

minusinfin119890minus12060121198872119889120601 (53)

Journal of Applied Mathematics 5

Then

119868 = intinfin

minusinfin119890minus1206012 (12058311987724119898 + 1198624)

11987711986721206014119889120601

minus1

2intinfin

minusinfin119890minus1206012 (1205831198776119898 + 1198623)

2

119877211986731206016119889120601

=12058311987724119898 + 1198624

1198771198672intinfin

minusinfin119890minus12060121206014119889120601

minus(1205831198776119898 + 1198623)

2

211987721198673intinfin

minusinfin119890minus12060121206016119889120601

=12058311987724119898 + 1198624

1198771198672sdot3radic120587

4minus(1205831198776119898 + 1198623)

2

11987721198673sdot15radic120587

16

(54)

Finally we obtain the approximation formula

119878120573119903 (119899119898) asymp119899

119898

119890120583119877(119890119877 minus 1)119898

2120573119898minus119899119877119899radic120587119898119877119867[1 +

119868

119898119877radic120587] (55)

In view of Lemmas 4 and 5 and (33) the previousasymptotic approximation is valid for 119898 ge (14)119899120583 Theformula obtained in the previous discussion is formally statedin the following theorem

Theorem 6 The formula

119878120573119903 (119899119898) sim 119876 [radic120587 +119868

119898119877] (56)

behaves as an asymptotic approximation for the generalizedStirling numbers of the second kind 119878120573119903(119899119898) with positiveintegral parameters for 119898 gt (14)119899120583 119899 gt 4 and such that119899 minus 119898 rarr infin as 119899 rarr infin

Table 1 displays the exact values and approximate valuesfor 119899 = 100 119903 = 4 120573 = 7 The exact values are obtainedusing recurrence formula while the approximate values areobtained using Theorem 6 In view of Lemma 5 the formulais valid for119898 ge 15

Values on the table affirm that the asymptotic formula inTheorem 6 gives a good approximation for 11987874(100119898)when119898 ge 15

4 Asymptotic Formula with Real Parameters

Recently the (119903 120573)-Stirling numbers for complex argumentswere defined in [15] parallel to the definition of Flajolet andProdinger as

119878120573119903 (119909 119910) =119909

1199101205731199102120587119894intH

119890119903119911(119890120573119911minus 1)119910 119889119911

119911119909+1 (57)

where 119909 = Γ(119909 + 1) and H is a Hankel contour thatstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeI119911 gt 0

Table 1

Exact value Approximate value Relative error11987874(100 5) 5685 times 10152 6335 times 10152 011428

11987874(100 10) 7728 times 10171

8169 times 10171

005713

11987874(100 15) 8411 times 10178 8731 times 10178 003804

11987874(100 30) 5604 times 10174 5706 times 10174 001824

11987874(100 60) 7399 times 10122

7446 times 10122

000641

11987874(100 80) 1275 times 1070

1279 times 1070

000263

11987874(100 90) 2208 times 1038 2211 times 1038 000123

Note that by change of variable say 119908 = 120573119911 we can express119878120573119903(119909 119910) as follows

119878120573119903 (119909 119910) = 120573119909minus119910

119909 +119903

120573

119910 +119903

120573

119903120573

(58)

where

119909 + 119903

119910 + 119903119903

=119909

1199102120587119894intH

119890119903119911(119890119911minus 1)119910 119889119911

119911119909+1 (59)

The numbers 119909+119903119910+119903 119903 are certain generalization of 119903-Stirling numbers of the second kind [8] in which the param-eters involved are complex numbers These numbers satisfythe following properties

Theorem 7 For nonnegative real number 119903 one has

119909 + 119903

119910 + 119903119903

= 119909 + 119903 minus 1

119910 + 119903 minus 1119903

+ (119910 + 119903) 119909 + 119903 minus 1

119910 + 119903119903

(60)

Theorem 8 For nonnegative real numbers 119903 and 120573 and 119896 isin None has

119909 + 119903

119896 + 119903119903

=1

119896

119896

sum119895=0

(minus1)119896minus119895(119896

119895) (119895 + 119903)

119909 (61)

Furthermore when 119896 = 119910 a complex number

119909 + 119903

119910 + 119903119903

=1

119910

infin

sum119895=0

(minus1)119910minus119895(119910

119895) (119895 + 119903)

119909 (62)

The Bernoulli polynomial can be expressed using theexplicit formula in [16] and the first formula in Theorem 8with 119908 = 119903 and 119909 = 119899 as

119861119899 (119903) =

infin

sum119896=0

(minus1)119896119896

119896 + 1119899 + 119903

119896 + 119903119903

(63)

Moreover it is known that for 119899 = 1 2 the Hurwitzzeta function 120577(1 minus 119899 119909) = minus119861119899(119909)119899 Note that as 119910 rarr 0(119910 minus 1) sim 1119910 By Cauchyrsquos integral formula

119899 + 119903

119910 + 119903 minus 1119903

sim 119910119899 [119911119899+1]119890119903119911

119890119911 minus 1= 119910119861119899+1 (119903)

119899 + 1 (64)

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

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DifferentialEquations

International Journal of

Volume 2013

4 Journal of Applied Mathematics

The equation in (37) can be written in the from

119891 (119911 119877 120601) =

infin

sum119896=1

119886119896119911119896 (38)

where

119886119896 = 119863119896+2 (119877)(119894120601)119896+2

119877119867(119896+2)2 (39)

Note here that 119911 = (119898119877)minus12 is within the radius of

convergence of the series in (38) On the other hand theMaclaurin series of exp119891(119911 119877 120601) is

119890119891(119911119877120601)

=

infin

sum119896=0

119887119896 (119877 120601) 119911119896 (40)

where 1198872119896 is a polynomial in 120601 containing only even powers of120601 and 1198872119896+1 is a polynomial in 120601 containing only odd powersof 120601 Now (36) can be written in the form

119878120573119903 (119899119898) sim 119876int120572

minus120572119890minus1206012(

infin

sum119896=0

119887119896 (119877 120601) 119911119896)119889120601 (41)

where 119876 = 119860radic119898119877119867 and 119911 = (119898119877)minus12We write (41) as

119878120573119903 (119899119898) sim 119876[

119904minus1

sum119896=0

119911119896(int120572

minus120572119890minus1206012119887119896119889120601) + 119880119904] (42)

where

119880119904 =

infin

sum119896=119904

(int120572

minus120572119890minus1206012119887119896119889120601) 119911

119896 (43)

In view of Lemma 2119867 ge 14 hence

120572 = (119898119877119867)12120598 = (119898119877119867)

12(119898119877)minus38

ge(119898119877)18

2 (44)

Since 119887119896 is a polynomial in 120601 we can replace 120572 with infinin (41) Following the discussion in [3] it can be shown that119880119904 = 119874(119911

119904) hence we have the asymptotic formula

119878120573119903 (119899119898) sim 119876[

infin

sum119896=0

(intinfin

minusinfin119890minus12060121198872119896119889120601) (119898119877)

minus119896] (45)

Note that

intinfin

minusinfin119890minus12060121198872119896+1119889120601 = 0 (46)

This is why no odd subscript of 119887119896 appears in (45)An approximation is obtained by taking the first two

terms of the sum in (45) Thus

119878120573119903 (119899119898) asymp 119876[intinfin

minusinfin119890minus12060121198870119889120601 + (119898119877)

minus1intinfin

minusinfin119890minus12060121198872119889120601]

(47)

It can be computed that

1198870 = 1

intinfin

minusinfin119890minus1206012119889120601 = radic120587

1198872 = 1198862 (119877 120601) +1

2[1198861 (119877 120601)]

2

1198862 = 1198634 (119877)(119894120601)4

1198771198672

1198634 (119877) =120583119877

24119898+ 1198624 (119877)

(48)

where

1198624 (119877) =1

24[119877 + (119877 minus 7119877

2+ 61198773minus 1198774) (119890119877minus 1)minus1]

+1

24[(18119877

3minus 71198772minus 71198774) (119890119877minus 1)minus2

+ (121198773minus 12119877

4) (119890119877minus 1)minus3]

minus1

24[61198774(119890119877minus 1)minus4]

(49)

while

1198861 (119877 120601) = 1198633(119894120601)3

11987711986732= (

120583119877

3119898+ 1198623)

(119894120601)3

11987711986732 (50)

where

1198623 =1

6[119877 + (119877 minus 3119877

2+ 1198773) (119890119877minus 1)minus1

+ (31198773minus 31198772) (119890119877minus 1)minus2]

+1

6[21198773(119890119877minus 1)minus3]

(51)

Thus 1198872 can be written as follows

1198872 = [120583119877

24119898+ 1198624]

1206014

1198771198672minus1

2(120583119877

6119898+ 1198623)

2 1206016

11987721198673 (52)

Let

119868 = intinfin

minusinfin119890minus12060121198872119889120601 (53)

Journal of Applied Mathematics 5

Then

119868 = intinfin

minusinfin119890minus1206012 (12058311987724119898 + 1198624)

11987711986721206014119889120601

minus1

2intinfin

minusinfin119890minus1206012 (1205831198776119898 + 1198623)

2

119877211986731206016119889120601

=12058311987724119898 + 1198624

1198771198672intinfin

minusinfin119890minus12060121206014119889120601

minus(1205831198776119898 + 1198623)

2

211987721198673intinfin

minusinfin119890minus12060121206016119889120601

=12058311987724119898 + 1198624

1198771198672sdot3radic120587

4minus(1205831198776119898 + 1198623)

2

11987721198673sdot15radic120587

16

(54)

Finally we obtain the approximation formula

119878120573119903 (119899119898) asymp119899

119898

119890120583119877(119890119877 minus 1)119898

2120573119898minus119899119877119899radic120587119898119877119867[1 +

119868

119898119877radic120587] (55)

In view of Lemmas 4 and 5 and (33) the previousasymptotic approximation is valid for 119898 ge (14)119899120583 Theformula obtained in the previous discussion is formally statedin the following theorem

Theorem 6 The formula

119878120573119903 (119899119898) sim 119876 [radic120587 +119868

119898119877] (56)

behaves as an asymptotic approximation for the generalizedStirling numbers of the second kind 119878120573119903(119899119898) with positiveintegral parameters for 119898 gt (14)119899120583 119899 gt 4 and such that119899 minus 119898 rarr infin as 119899 rarr infin

Table 1 displays the exact values and approximate valuesfor 119899 = 100 119903 = 4 120573 = 7 The exact values are obtainedusing recurrence formula while the approximate values areobtained using Theorem 6 In view of Lemma 5 the formulais valid for119898 ge 15

Values on the table affirm that the asymptotic formula inTheorem 6 gives a good approximation for 11987874(100119898)when119898 ge 15

4 Asymptotic Formula with Real Parameters

Recently the (119903 120573)-Stirling numbers for complex argumentswere defined in [15] parallel to the definition of Flajolet andProdinger as

119878120573119903 (119909 119910) =119909

1199101205731199102120587119894intH

119890119903119911(119890120573119911minus 1)119910 119889119911

119911119909+1 (57)

where 119909 = Γ(119909 + 1) and H is a Hankel contour thatstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeI119911 gt 0

Table 1

Exact value Approximate value Relative error11987874(100 5) 5685 times 10152 6335 times 10152 011428

11987874(100 10) 7728 times 10171

8169 times 10171

005713

11987874(100 15) 8411 times 10178 8731 times 10178 003804

11987874(100 30) 5604 times 10174 5706 times 10174 001824

11987874(100 60) 7399 times 10122

7446 times 10122

000641

11987874(100 80) 1275 times 1070

1279 times 1070

000263

11987874(100 90) 2208 times 1038 2211 times 1038 000123

Note that by change of variable say 119908 = 120573119911 we can express119878120573119903(119909 119910) as follows

119878120573119903 (119909 119910) = 120573119909minus119910

119909 +119903

120573

119910 +119903

120573

119903120573

(58)

where

119909 + 119903

119910 + 119903119903

=119909

1199102120587119894intH

119890119903119911(119890119911minus 1)119910 119889119911

119911119909+1 (59)

The numbers 119909+119903119910+119903 119903 are certain generalization of 119903-Stirling numbers of the second kind [8] in which the param-eters involved are complex numbers These numbers satisfythe following properties

Theorem 7 For nonnegative real number 119903 one has

119909 + 119903

119910 + 119903119903

= 119909 + 119903 minus 1

119910 + 119903 minus 1119903

+ (119910 + 119903) 119909 + 119903 minus 1

119910 + 119903119903

(60)

Theorem 8 For nonnegative real numbers 119903 and 120573 and 119896 isin None has

119909 + 119903

119896 + 119903119903

=1

119896

119896

sum119895=0

(minus1)119896minus119895(119896

119895) (119895 + 119903)

119909 (61)

Furthermore when 119896 = 119910 a complex number

119909 + 119903

119910 + 119903119903

=1

119910

infin

sum119895=0

(minus1)119910minus119895(119910

119895) (119895 + 119903)

119909 (62)

The Bernoulli polynomial can be expressed using theexplicit formula in [16] and the first formula in Theorem 8with 119908 = 119903 and 119909 = 119899 as

119861119899 (119903) =

infin

sum119896=0

(minus1)119896119896

119896 + 1119899 + 119903

119896 + 119903119903

(63)

Moreover it is known that for 119899 = 1 2 the Hurwitzzeta function 120577(1 minus 119899 119909) = minus119861119899(119909)119899 Note that as 119910 rarr 0(119910 minus 1) sim 1119910 By Cauchyrsquos integral formula

119899 + 119903

119910 + 119903 minus 1119903

sim 119910119899 [119911119899+1]119890119903119911

119890119911 minus 1= 119910119861119899+1 (119903)

119899 + 1 (64)

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

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Stochastic AnalysisInternational Journal of

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DifferentialEquations

International Journal of

Volume 2013

Journal of Applied Mathematics 5

Then

119868 = intinfin

minusinfin119890minus1206012 (12058311987724119898 + 1198624)

11987711986721206014119889120601

minus1

2intinfin

minusinfin119890minus1206012 (1205831198776119898 + 1198623)

2

119877211986731206016119889120601

=12058311987724119898 + 1198624

1198771198672intinfin

minusinfin119890minus12060121206014119889120601

minus(1205831198776119898 + 1198623)

2

211987721198673intinfin

minusinfin119890minus12060121206016119889120601

=12058311987724119898 + 1198624

1198771198672sdot3radic120587

4minus(1205831198776119898 + 1198623)

2

11987721198673sdot15radic120587

16

(54)

Finally we obtain the approximation formula

119878120573119903 (119899119898) asymp119899

119898

119890120583119877(119890119877 minus 1)119898

2120573119898minus119899119877119899radic120587119898119877119867[1 +

119868

119898119877radic120587] (55)

In view of Lemmas 4 and 5 and (33) the previousasymptotic approximation is valid for 119898 ge (14)119899120583 Theformula obtained in the previous discussion is formally statedin the following theorem

Theorem 6 The formula

119878120573119903 (119899119898) sim 119876 [radic120587 +119868

119898119877] (56)

behaves as an asymptotic approximation for the generalizedStirling numbers of the second kind 119878120573119903(119899119898) with positiveintegral parameters for 119898 gt (14)119899120583 119899 gt 4 and such that119899 minus 119898 rarr infin as 119899 rarr infin

Table 1 displays the exact values and approximate valuesfor 119899 = 100 119903 = 4 120573 = 7 The exact values are obtainedusing recurrence formula while the approximate values areobtained using Theorem 6 In view of Lemma 5 the formulais valid for119898 ge 15

Values on the table affirm that the asymptotic formula inTheorem 6 gives a good approximation for 11987874(100119898)when119898 ge 15

4 Asymptotic Formula with Real Parameters

Recently the (119903 120573)-Stirling numbers for complex argumentswere defined in [15] parallel to the definition of Flajolet andProdinger as

119878120573119903 (119909 119910) =119909

1199101205731199102120587119894intH

119890119903119911(119890120573119911minus 1)119910 119889119911

119911119909+1 (57)

where 119909 = Γ(119909 + 1) and H is a Hankel contour thatstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeI119911 gt 0

Table 1

Exact value Approximate value Relative error11987874(100 5) 5685 times 10152 6335 times 10152 011428

11987874(100 10) 7728 times 10171

8169 times 10171

005713

11987874(100 15) 8411 times 10178 8731 times 10178 003804

11987874(100 30) 5604 times 10174 5706 times 10174 001824

11987874(100 60) 7399 times 10122

7446 times 10122

000641

11987874(100 80) 1275 times 1070

1279 times 1070

000263

11987874(100 90) 2208 times 1038 2211 times 1038 000123

Note that by change of variable say 119908 = 120573119911 we can express119878120573119903(119909 119910) as follows

119878120573119903 (119909 119910) = 120573119909minus119910

119909 +119903

120573

119910 +119903

120573

119903120573

(58)

where

119909 + 119903

119910 + 119903119903

=119909

1199102120587119894intH

119890119903119911(119890119911minus 1)119910 119889119911

119911119909+1 (59)

The numbers 119909+119903119910+119903 119903 are certain generalization of 119903-Stirling numbers of the second kind [8] in which the param-eters involved are complex numbers These numbers satisfythe following properties

Theorem 7 For nonnegative real number 119903 one has

119909 + 119903

119910 + 119903119903

= 119909 + 119903 minus 1

119910 + 119903 minus 1119903

+ (119910 + 119903) 119909 + 119903 minus 1

119910 + 119903119903

(60)

Theorem 8 For nonnegative real numbers 119903 and 120573 and 119896 isin None has

119909 + 119903

119896 + 119903119903

=1

119896

119896

sum119895=0

(minus1)119896minus119895(119896

119895) (119895 + 119903)

119909 (61)

Furthermore when 119896 = 119910 a complex number

119909 + 119903

119910 + 119903119903

=1

119910

infin

sum119895=0

(minus1)119910minus119895(119910

119895) (119895 + 119903)

119909 (62)

The Bernoulli polynomial can be expressed using theexplicit formula in [16] and the first formula in Theorem 8with 119908 = 119903 and 119909 = 119899 as

119861119899 (119903) =

infin

sum119896=0

(minus1)119896119896

119896 + 1119899 + 119903

119896 + 119903119903

(63)

Moreover it is known that for 119899 = 1 2 the Hurwitzzeta function 120577(1 minus 119899 119909) = minus119861119899(119909)119899 Note that as 119910 rarr 0(119910 minus 1) sim 1119910 By Cauchyrsquos integral formula

119899 + 119903

119910 + 119903 minus 1119903

sim 119910119899 [119911119899+1]119890119903119911

119890119911 minus 1= 119910119861119899+1 (119903)

119899 + 1 (64)

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamicsin Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

DifferentialEquations

International Journal of

Volume 2013

6 Journal of Applied Mathematics

which further gives using a result in [17] the followingrelation

[119889

119889119910119899 + 119903

119910 + 119903119903

]119910=minus1

= minus120577 (minus119899 119903)

=119899

2120587119894intH

119890119903119911(119890119911minus 1)minus1 119889119911

119911119899+1

(65)

An asymptotic formula for 119903-Stirling numbers of thesecond kind was first considered by Corcino et al in [14]However the formula only holds for integral arguments Inthis section we are going to establish an asymptotic formulafor 119903-Stirling numbers of the second kind that will hold forreal arguments

Consider the integral representation in (59) where |I119911| lt2120587 To see the analysis of Moser-Wyman applies to 119903-Stirlingnumbers with real arguments 119909 and 119910 we deform the pathHinto the following contour 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 where

(1) 1198621 is the line I119911 = minus2120587 + 120575 120575 gt 0 andR119911 le 120598 120598 is asmall positive number

(2) 1198622 is the line segmentR119911 = 120598 going from 120598+119894(120575minus2120587)to the circle |119911| = 119877

(3) 1198625 and1198624 are the reflections in the real axis of 1198621 and1198622 respectively and

(4) 1198623 is the portion of the circle |119911| = 119877 meeting 1198622 and1198624

The new contour is 1198621 cup 1198622 cup 1198623 cup 1198624 cup 1198625 in thecounterclockwise sense This idea of deforming the contourH is also done in [11]

The integrals along 1198621 1198622 1198624 and 1198625 are seen to be

119874((2 + 120598)

119910

(2120587 minus 120575)119909) (66)

Itwill also be shown that these integrals go to 0 as119909 rarr infin

provided that 119909 minus 119910 ge 119909120572 where 0 lt 120572 lt 1 To see this weconsider 1198621 For the other contours the estimate can be seensimilarly

Note that for 119911 in 1198621 we can choose 120575 so that

[1 minus (119890R119911 cos (I119911))

minus1]2

lt 1 (67)

From this and the assumptions thatR119911 lt 120598 and 120598 lt 1 itfollows that

1003816100381610038161003816119890119911minus 11003816100381610038161003816 le 119890

R119911lt 120598 lt

2 + 120598

2 minus 120598lt 2 + 120598 (68)

Thus1003816100381610038161003816119890119911minus 11003816100381610038161003816119910lt (2 + 120598)

119910 (69)

Consequently

100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le int1198621

(2 + 120598)119910

(2120587 minus 120575)119909+1|119889119911| (70)

where int1198621|119889119911| is the length of 1198621 With 1198621 the horizontal line

I119911 = minus2120587 + 120575 the length of 1198621 is a linear function of the realpart of 119911 given by

119897 (1198621) = lim119905rarrinfin

(120598 minus 119905) (71)

Hence we have100381610038161003816100381610038161003816100381610038161003816int1198621

(119890119911 minus 1)119910

119911119909+1119889119911

100381610038161003816100381610038161003816100381610038161003816le

(2 + 120598)119910

(2120587 minus 120575)119909+1

lim119905rarrinfin

(120598 minus 119905)

ltlim119905rarrinfin (120598 minus 119905)(2120587 minus 120575)

119909minus119910

(72)

The last inequality follows from the fact that 2+120598 lt 2120587minus120575With the condition that 119909 minus 119910 ge 119909120572 0 lt 120572 lt 1 and the factthat 2120587 minus 120575 gt 119890 it follows that the integral along 1198621 goes to 0as 119909 rarr infin Thus we have

119909 + 119903

119910 + 119903119903

sim119909

119910

1

2120587119894int1198623

119890119903119911(119890119911 minus 1)119910

119911119909+1119889119911 (73)

Using the method of Moser and Wyman [3] we obtainthe following asymptotic formula

Theorem 9 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (74)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 119909120572with 0 lt 120572 lt 1 where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (75)

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (76)

as a function of 119908

Chelluri et al [11] has made a modification of Moser andWyman formula and analysis Using this analysis of Chelluriwe can restate Theorem 9 as follows

Theorem 10 The 119903-Stirling numbers of the second kind withreal arguments 119909 and 119910 have the following asymptotic formula

119909 + 119903

119910 + 119903119903

=119909

119910

119890119903119877(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (77)

valid for 119909 minus 119910 rarr infin as 119909 rarr infin provided that 119909 minus 119910 ge 11990913where

119867 =119903

2119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2 (78)

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamicsin Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

DifferentialEquations

International Journal of

Volume 2013

Journal of Applied Mathematics 7

and 119877 is the unique positive solution to the equation

119908[119903 minus 119910(1 minus 119890minus119908)minus1] minus 119909 = 0 (79)

as a function of 119908

As noted in (58) the (119903 120573)-Stirling numbers can beobtained using the 119903-Stirling numbers of the second kindfor complex arguments 119909 119910 119903 and 120573 This implies usingTheorems 9 and 10 the following asymptotic formulas whichagree with the formula in (55) for the integral values of 119909 and119910

Corollary 11 The (119903 120573)-Stirling numbers with real arguments119909 and 119910 have the following asymptotic formulas

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119910)] (80)

provided that 119909 minus 119910 ge 119909120572 with 0 lt 120572 lt 1 and

119878120573119903 (119909 119910) =120573119909minus119910119909

119910

119890119903119877120573(119890119877 minus 1)119910

2119877119909radic120587119910119877119867[1 + 119874(

1

119909)] (81)

provided that 119909 minus 119910 ge 11990913 where

119867 =119903

2120573119910+119890119877 (119890119877 minus 119877 minus 1)

2(119890119877 minus 1)2

(82)

and 119877 is the unique positive solution to the equation

119908[119903

120573minus 119910(1 minus 119890

minus119908)minus1] minus 119909 = 0 (83)

as a function of 119908

Acknowledgment

The authors would like to thank the referee for reading andevaluating the paper

References

[1] N M Temme ldquoAsymptotic estimates of Stirling numbersrdquoStudies in AppliedMathematics vol 89 no 3 pp 233ndash243 1993

[2] L Moser and M Wyman ldquoAsymptotic development of theStirling numbers of the first kindrdquo Journal of the LondonMathematical Society vol 33 pp 133ndash146 1958

[3] LMoser andMWyman ldquoStirling numbers of the second kindrdquoDuke Mathematical Journal vol 25 pp 29ndash43 1957

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the (119903 120573)-Stirling numbersrdquoArsCombinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] L C Hsu and P J S Shuie ldquoA unified approach to generalizedstirling numbersrdquoAdvances in AppliedMathematics vol 20 no3 pp 366ndash384 1998

[8] A Z Broder ldquoThe 119903-stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[9] R B Corcino and R Aldema ldquoSome combinatorial andstatistical applications of (119903 120573)-Stirling numbersrdquo MatimyasMatematika vol 25 no 1 pp 19ndash29 2002

[10] R B Corcino and C B Corcino ldquoOn the maximum ofgeneralized Stirling numbersrdquoUtilitas Mathematica vol 86 pp241ndash256 2011

[11] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

[12] M A R P Vega and C B Corcino ldquoAn asymptotic formulafor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 73 pp 129ndash141 2007

[13] M A R P Vega and C B Corcino ldquoMore asymptotic formulasfor the generalized Stirling numbers of the first kindrdquo UtilitasMathematica vol 75 pp 259ndash272 2008

[14] C B Corcino L C Hsu and E L Tan ldquoAsymptotic approx-imations of 119903-stirling numbersrdquo Approximation Theory and ItsApplications vol 15 no 3 pp 13ndash25 1999

[15] R B Corcino C B Corcino and M B Montero ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[16] F W J Olver D Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress Washington DC USA 2010

[17] K N Boyadzhiev ldquoEvaluation of series with Hurwitz and Lerchzeta function coefficients by using Hankel contour integralsrdquoAppliedMathematics and Computation vol 186 no 2 pp 1559ndash1571 2007

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Mathematical Problems in Engineering

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Abstract and Applied Analysis

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences

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ISRN Geometry

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamicsin Nature and Society

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Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

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ProbabilityandStatistics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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Journal ofApplied Mathematics

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DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

DifferentialEquations

International Journal of

Volume 2013

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamicsin Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

DifferentialEquations

International Journal of

Volume 2013