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Hindawi Publishing CorporationJournal of Complex AnalysisVolume 2013 Article ID 538592 7 pageshttpdxdoiorg1011552013538592

Research ArticleSome New Explicit Values of Quotients of Theta-Function 120601(119902)and Applications to Ramanujanrsquos Continued Fractions

Nipen Saikia

Department of Mathematics Rajiv Gandhi University Rono Hills Doimukh Arunachal Pradesh 791112 India

Correspondence should be addressed to Nipen Saikia nipennakyahoocom

Received 14 November 2012 Accepted 23 January 2013

Academic Editor Nikolai Tarkhanov

Copyright copy 2013 Nipen Saikia This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We find some new explicit values of the parameter ℎ119896119899 for positive real numbers k and n involving Ramanujanrsquos theta-function 120601(q)and give some applications of these new values for the explicit evaluations of Ramanujanrsquos continued fractions In the process wealso establish two new identities for 120601(q) by using modular equations

1 Introduction

For 119902 = 1198902120587119894119911 Im(119911) gt 0 define Ramanujanrsquos theta-function

120601(119902) as

120601 (119902) =

infin

sum

119899=minusinfin

1199021198992

= 1205993 (0 2119911) (1)

where 1205993 [1 page 464] is one of the classical theta-functionsIn his notebook [2 volume I page 248] Ramanujan

recorded several explicit values of theta-functions120601(119902) and itsquotients which are proved by Berndt andChan [3]They alsofound some new explicit values An account of these can alsobe found in Berndtrsquos book [4] Yi [5] also evaluatedmany newvalues of 120601(119902) by finding explicit values of the parameters ℎ119896119899and ℎ1015840

119896119899for positive real numbers 119896 and 119899 which are defined

by

ℎ119896119899 =120601 (119902)

11989614120601 (119902119896) 119902 = 119890

minus120587radic119899119896

ℎ1015840

119896119899=

120601 (minus119902)

11989614120601 (minus119902119896) 119902 = 119890

minus2120587radic119899119896

(2)

Yi [5] established several properties of these parameters andfound their explicit values by appealing to transformationformulas and theta-function identities for 120601(119902) RecentlySaikia [6] found many new explicit values of quotients of120601(119902) by finding explicit values of the parameter119860119899 which is a

particular case of the parameter ℎ119896119899 where 119896 = 4 Saikia [6]also established some new theta-function identities for 120601(119902)

In the sequel of the previous work in this paper wefind some new explicit values of the parameters ℎ4119899 andℎ2119899 which are particular cases of the parameter ℎ119896119899 byusing some properties of ℎ119896119899 established in [5] and twonew theta-function identities for 120601(119902) In addition we givesome applications of these new values of ℎ119896119899 for the explicitevaluations of Ramanujanrsquos continued fractions 119888(119902) and119870(119902)defined by

119888 (119902) =1

1+

2119902

1 minus 1199022+

1199022(1 + 119902

2)2

1 minus 1199026+

1199024(1 + 119902

4)2

1 minus 11990210+sdotsdotsdot

10038161003816100381610038161199021003816100381610038161003816 lt 1

(3)

119870(119902) =11990212

1 + 119902+

1199022

1 + 1199023+

1199024

1 + 1199025+sdotsdotsdot

10038161003816100381610038161199021003816100381610038161003816 lt 1

(4)

The continued fraction 119888(119902) is studied by Adiga and Anitha[7] For explicit evaluations of 119888(119902) see [6] The continuedfraction 119870(119902) is called Ramanujan-Gollnitz-Gordon contin-ued fraction [4 page 50] For further references on 119870(119902) see[8 9]

In Section 2 we record some preliminary resultsSection 3 is devoted to prove two new identities for theta-function 120601(119902) In Section 4 we find some new explicit values

2 Journal of Complex Analysis

of the parameter ℎ4119899 In Section 5 we evaluate some newvalues of the parameter ℎ2119899 Finally in Section 6 we giveapplications of these values of ℎ4119899 and ℎ2119899 for the explicitevaluations of Ramanujanrsquos continued fractions 119888(119902) and119870(119902)

We end the introduction by defining Ramanujanrsquos mod-ular equation The complete elliptic integral of the first kind119870(119896) is defined by

119870 (119896) = int

1205872

0

119889120601

radic1 minus 1198962sin2120601=120587

2

infin

sum

119899=0

(12)2

119899

(119899)21198962119899

=120587

221198651 (

1

21

2 1 1198962)

(5)

where 0 lt 119896 lt 1 21198651 denotes the ordinary or Gaussianhypergeometric function and

(119886)119899 = 119886 (119886 + 1) (119886 + 2) sdot sdot sdot (119886 + 119899 minus 1) (6)

The number 119896 is called the modulus of 119870 and 1198961015840

=

radic1 minus 1198962 is called the complementary modulus Let 1198701198701015840 119871and 119871

1015840 denote the complete elliptic integrals of the firstkind associated with the moduli 119896 1198961015840 119897 and 1198971015840 respectivelySuppose that the equality

1198991198701015840

119870=1198711015840

119871(7)

holds for some positive integer 119899 Then a modular equationof degree 119899 is a relation between the moduli 119896 and 119897 which isimplied by (7)

If we set

119902 = exp(minus1205871198701015840

119870) 119902

1015840= exp(minus120587119871

1015840

119871) (8)

we see that (7) is equivalent to the relation 119902119899= 1199021015840 Thus

a modular equation can be viewed as an identity involvingtheta-functions at the arguments 119902 and 119902

119899 Ramanujanrecorded his modular equations in terms of 120572 and 120573 where120572 = 119896

2 and 120573 = 1198972 We say that 120573 has degree 119899 over 120572 The

multiplier 119898 connecting 120572 and 120573 is defined by 119898 = 1199111119911119899

where 119911119903 = 1206012(119902119903)

2 Preliminary Results

Lemma 1 (see [5 page 385 Theorem 22]) For positive realnumbers 119896 and 119899

(i) ℎ1198961 = 1(ii) ℎ1198961119899 = 1ℎ119896119899

Lemma 2 (see [5 page 387 Corollary 26]) For positive realnumbers 119896 and 119899

ℎ1198962 119899 = ℎ119896119899119896ℎ119896119899119896 (9)

Lemma 3 (see [5 page 392 Theorem 46]) One has

radic2(ℎ22119899ℎ24119899 +1

ℎ22119899ℎ24119899

) =ℎ24119899

ℎ2119899

+ 2 (10)

for any positive real number n

Lemma 4 (see [10 page 122 Entry 10 (i) and (v)]) One has

(i) 120601(119902) = radic119911

(ii) 120601(1199024) = radic119911(1 + (1 minus 120572)14))2

Lemma 5 (see [10 page 280-281 Entry 13(vii)]) If 120573 hasdegree 5 over 120572 then

(1205721205733)18

+ (1 minus 120572)(1 minus 120573)318

= (1205723120573)18

+ (1 minus 120572)3(1 minus 120573)

18

(11)

Lemma 6 (see [10 page 314 Entry 19(i)]) If 120573 has degree 7over 120572 then

(120572120573)18

+ (1 minus 120572)(1 minus 120573)18

= 1 (12)

3 Two New Identities for 120601(119902)

In this section we prove two new identities for theta-function 120601(119902) by using Ramanujanrsquos modular equations andtransformation formulas

Theorem 7 If 119875 = 120601(119902)120601(1199024) 119876 = 120601(119902

5)120601(11990220) then

1198756minus 256119875119876 + 640119875

2119876 minus 640119875

3119876

+ 3201198754119876 minus 70119875

5119876 + 640119875119876

2minus 1600119875

21198762

+ 160011987531198762minus 785119875

41198762+ 160119875

51198762minus 640119875119876

3

+ 160011987521198763minus 1620119875

31198763+ 800119875

41198763minus 160119875

51198763

+ 3201198751198764minus 785119875

21198764+ 800119875

31198764minus 400119875

41198764

+ 8011987551198764minus 70119875119876

5+ 160119875

21198765minus 160119875

31198765+ 80119875

41198765

minus 1611987551198765+ 1198766= 0

(13)

Proof Transcribing 119875 and 119876 using Lemma 4(i) and (iv) andthen simplifying we get

(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (14)

where 120573 has degree 5 over 120572Equivalently

120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(15)

Journal of Complex Analysis 3

Now by Lemma 5 we have

(1205721205733)18

minus (1205723120573)18

= (1 minus 120572)3(1 minus 120573)

18

minus (1 minus 120572) (1 minus 120573)318

(16)

Squaring (16) and simplifying we arrive at

(1205721205733)14

+ (1205723120573)14

= 119909 + 2(120572120573)12 (17)

where

119909 = (1 minus 120572)3(1 minus 120573)

14

+ (1 minus 120572) (1 minus 120573)314

minus 2(1 minus 120572) (1 minus 120573)12

(18)

Squaring (17) and simplifying we obtain

(1205721205733)12

+ (1205723120573)12

= 119910 + 4119909(120572120573)12 (19)

where 119910 = 1199092+ 2120572120573


1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573 = 8119909119910(120572120573)

12 (20)

Again squaring (20) we obtain

(1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573)2

minus 6411990921199102120572120573 = 0

(21)

Now employing (14) and (15) and factorizing using Mathe-matica we deduce that

119891 (119875119876) 119892 (119875 119876) ℎ (119875 119876) = 0 (22)

where

119891 (119875119876) = (119875 minus 119876)4

119892 (119875 119876) = 1198756minus 256119875119876 + 640119875

2119876

minus 6401198753119876 + 320119875

4119876 minus 70119875

5119876 + 640119875119876

2

minus 160011987521198762+ 1600119875

31198762minus 785119875

41198762+ 160119875

51198762

minus 6401198751198763+ 1600119875

21198763minus 1620119875

31198763+ 800119875

41198763

minus 16011987551198763+ 320119875119876

4minus 785119875

21198764+ 800119875

31198764

minus 40011987541198764+ 80119875

51198764minus 70119875119876

5+ 160119875

21198765

minus 16011987531198765+ 80119875

41198765minus 16119875

51198765+ 1198766

ℎ (119875 119876)

= 1611987510minus 96119875

9119876 minus 32119875

10119876 + 240119875

81198762+ 192119875

91198762

+ 24119875101198762minus 320119875

71198763minus 480119875

81198763minus 144119875

91198763

minus 8119875101198763+ 256119875

61198764+ 608119875

71198764+ 384119875

81198764

+ 4011987591198764+ 119875101198764minus 448119875

51198765+ 352119875

61198765

minus 126411987571198765+ 248119875

81198765minus 70119875

91198765+ 256119875

41198766

+ 35211987551198766minus 880119875

61198766+ 1640119875

71198766minus 785119875

81198766

+ 16011987591198766minus 320119875

31198767+ 608119875

41198767minus 1264119875

51198767

+ 164011987561198767minus 1620119875

71198767+ 800119875

81198767minus 160119875

91198767

+ 24011987521198768minus 480119875

31198768+ 384119875

41198768+ 248119875

51198768

minus 78511987561198768+ 800119875

71198768minus 400119875

81198768+ 80119875

91198768

minus 961198751198769+ 192119875

21198769minus 144119875

31198769+ 40119875

41198769

minus 7011987551198769+ 160119875

61198769minus 160119875

71198769+ 80119875

81198769

minus 1611987591198769+ 16119876

10minus 32119875119876

10+ 24119875

211987610

minus 8119875311987610+ 119875411987610

(23)

By examining the behavior of the first factor 119891(119875119876) andthe last factor ℎ(119875 119876) of the left-hand side of (22) near 119902 =

0 it can be seen that there is a neighborhood about theorigin where these factors are not zero Then the secondfactor 119892(119875 119876) is zero in this neighborhood By the identitytheorem this factor is identically zero Hence we completethe proof

Theorem 8 If 119875 = 120601(119902)120601(1199024) 119876 = 120601(119902

7)120601(11990228) then

1198758minus 4096119875119876 + 14336119875

2119876 minus 21504119875

3119876 + 17920119875

4119876

minus 87361198755119876 + 2352119875

6119876 minus 280119875

7119876 + 14336119875119876

2

minus 5196811987521198762+ 80640119875

31198762minus 69440119875

41198762+ 35056119875

51198762

minus 977211987561198762+ 1176119875

71198762minus 21504119875119876

3+ 80640119875

21198763

minus 12947211987531198763+ 115360119875

41198763minus 60424119875

51198763+ 17528119875

61198763

minus 218411987571198763+ 17920119875119876

4minus 69440119875

21198764+ 115360119875

31198764

minus 10633011987541198764+ 57680119875

51198764minus 17360119875

61198764+ 2240119875

71198764

minus 87361198751198765+ 35056119875

21198765minus 60424119875

31198765+ 57680119875

41198765

minus 3236811987551198765+ 10080119875

61198765minus 1344119875

71198765+ 2352119875119876

6

minus 977211987521198766+ 17528119875

31198766minus 17360119875

41198766+ 10080119875

51198766

minus 324811987561198766+ 448119875

71198766minus 280119875119876

7+ 1176119875

21198767

minus 218411987531198767+ 2240119875

41198767minus 1344119875

51198767+ 448119875

61198767

minus 6411987571198767+ 1198768= 0

(24)


Proof Transcribing 119875 and 119876 using Lemma 4(i) and (iv) andthen simplifying we obtain

(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (25)


120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(26)


(1 minus 120572) (1 minus 120573)18

= 1 minus (120572120573)18 (27)


119909 minus (120572120573)14

= minus2(120572120573)18 (28)

where 119909 = (1 minus 120572)(1 minus 120573)14

minus 1Squaring (28) and simplifying we obtain

1199092+ (120572120573)

12= (4 + 2119909) (120572120573)

14 (29)


1199094+ 120572120573 = ((4 + 2119909)

2minus 21199092) (120572120573)

12 (30)

Again squaring (30) we arrive at

(1199094+ 120572120573)

2

= ((4 + 2119909)2minus 21199092)2

(120572120573) (31)

Employing (25) and (26) in (31) and simplifying with the helpofMathematica we complete the proof

4 New Values of ℎ4119899

In this section we find some new values of ℎ4119899 and ℎ2119899 byusing theta-function identities proved in Section 3 and theproperties of ℎ119896119899 listed in Lemmas 1 and 2 We begin withfollowing remarks

Remark 9 The values of ℎ119896119899 are real and ℎ119896119899 lt 1 for all119899 gt 1 if 119896 gt 1 We also note that the values of ℎ119896119899 decreaseas 119899 increases when 119896 gt 1 In view of this in the followingtheorem we have ℎ44 gt ℎ45 gt ℎ47 gt ℎ48 gt ℎ425 gt ℎ449where ℎ44 and ℎ48 are evaluated in [6 Theorem 43] Yi [5]also evaluated the value of ℎ44

Theorem 10 One has

(i) ℎ45 = (radic22 + 10radic5 minus radic2 minus radic10 minus

radic(radic2 + radic10 minus radic22 + 10radic5)

2

minus 4)2

(ii) ℎ415 = (radic22 + 10radic5 minus radic2 minus radic10 +


2

minus 4)2

(iii) ℎ425 = 114 minus 80radic2 + radic25965 minus 18360radic2 minus

radic(114 minus 80radic2 + radic25965 minus 18360radic2)

2

minus 1

(iv) ℎ4125 = 114 minus 80radic2 + radic25965 minus 18360radic2 +


2

minus 1

(v) ℎ47 = (12 minus 7radic2 minus radic7(34 minus 24radic2))2

(vi) ℎ417 = (12 minus 7radic2 + radic7(34 minus 24radic2))2

(vii) ℎ449 = 119886 + radic119887 minus radicminus1 + (119886 + radic119887)2

(viii) ℎ4149 = 119886 + radic119887 + radicminus1 + (119886 + radic119887)2

where 119886 = 1317 minus 931radic2 + radic3470684 minus 2454144radic2

and 119887 = 6940535 minus 4907700radic2 +

1306133196radic7(123953 minus 87648radic2) minus

923575640radic14(123953 minus 87648radic2)

Proof For (i) and (ii) setting 119896 = 4 and employing thedefinition of ℎ119896119899 in Theorem 7 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ425119899 (32)

Setting 119899 = 15 in (32) applying to (13) and then simplifyingusing Lemma 1(ii) we obtain

(ℎ6

45+ ℎminus6

45) minus 70 (ℎ

4

45+ ℎminus4

45) + 320radic2 (ℎ

3

45+ ℎminus3

45)

minus 1425 (ℎ2

45+ ℎminus2

45) + 1920radic2 (ℎ45 + ℎ

minus1

45) minus 3348 = 0

(33)

Equivalently

1198606minus 76119860

4+ 320radic2119860

3minus 1136119860

2+ 960radic2119860 minus 640 = 0

(34)

where

119860 = ℎ45 + ℎminus1

45 (35)

Solving (34) for 119860 and noting that 119860 has positive real valuegreater than 1 we obtain

119860 = minusradic2 minus radic10 + radic22 + 10radic5 (36)

Invoking (36) in (35) solving for ℎ45 and using the fact inRemark 9 we complete the proof of (i) Noting ℎ415 = 1ℎ45from Lemma 1(ii) we arrive at (ii)

For proofs of (iii) and (iv) we set 119899 = 1 in (32) applyingto (13) and then simplifying using Lemma 1(i) we arrive at

(ℎ2

425+ ℎminus2

425) minus (456 minus 320radic2) (ℎ425 + ℎ

minus1

425)

minus (674 minus 840radic2) = 0

(37)

Equivalently

1198612+ 8 (minus57 + 40radic2) 119861 + (minus674 + 840radic2) = 0 (38)


where

119861 = ℎ425 + ℎminus1

425 (39)


119861 = 2 (114 minus 80radic2 + radic5 (5193 minus 3672radic2)) (40)

Invoking (40) in (39) solving for ℎ425 and using the fact inRemark 9 we complete the proof of (iii) Noting ℎ4125 =

1ℎ425 from Lemma 1(iv) we prove (ii)To prove (v) and (vi) applying the definition of ℎ4119899 in

Theorem 8 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ449119899 (41)

Setting 119899 = 17 in (41) applying in (24) and then simplifyingusing Lemma 1(ii) we arrive at

ℎ16

47minus 280ℎ

14

47+ 2352radic2ℎ

13

47minus 18508ℎ

12

47

+ 44016radic2ℎ11

47minus 140616ℎ

10

47+ 159264radic2ℎ

9

47

minus 262810ℎ8

47+ 159264radic2ℎ

7

47minus 140616ℎ

6

47

+ 44016radic2ℎ5

47minus 18508ℎ

4

47+ 2352radic2ℎ

3

47

minus 280ℎ2

47+ 1 = 0

(42)

Dividing (42) by ℎ847

and simplifying we get

(ℎ8

47+ ℎminus8

47) minus 280 (ℎ

6

47+ ℎminus6

47) + 2352radic2 (ℎ

5

47+ ℎminus5

47)

minus 18508 (ℎ4

47+ ℎminus4

47) + 44016radic2 (ℎ

3

47+ ℎminus3

47)

minus 140616 (ℎ2

47+ ℎminus2

47) + 159264radic2 (ℎ47 + ℎ

minus1

47)

minus 262810 = 0

(43)

Equivalently

1198718minus 288119871

6+ 2352radic2119871

5minus 16808119871

4+ 32256radic2119871

3minus 69120119871

2

+ 38976radic2119871 minus 18032 = 0

(44)

where

119871 = ℎ47 + ℎminus1

47 (45)

Solving (44) by usingMathematica and noting that 119871 has pos-itive real value greater that 1 satisfying the fact in Remark 9we obtain

119871 = 12 minus 7radic2 (46)

Employing (46) in (45) solving for ℎ47 and using the factin Remark 9 we complete the proof of (v) Noting ℎ417 =1ℎ47 we arrive at (vi)

For proofs of (vii) and (viii) setting 119899 = 1 in (41) applyingin (24) and simplifying using Lemma 1(ii) we arrive at

ℎ4

449+ ℎminus4

449+ 4 (minus1317 + 931radic2) (ℎ

3

449+ ℎminus3

449)

+ 28 (minus2053 + 1452radic2) (ℎ2

449+ ℎminus2

449)

+ 56 (minus2409 + 1703radic2) (ℎ449 + ℎminus1

449)

+ 210 (minus837 + 592radic2) = 0

(47)

Equivalently

1198634+ 8 (minus1317 + 931radic2)119863

3+ 16 (minus3593 + 2541radic2)119863

2

+ 64 (minus1614 + 1141radic2)119863 + 128 (minus475 + 336radic2) = 0

(48)

where

119863 = ℎ449 + ℎminus1

449 (49)


119863 = 2 (119886 + radic119887) (50)


and 119887 = 6940535 minus 4907700radic2 +


923575640radic14(123953 minus 87648radic2)Invoking (50) in (49) solving for ℎ449 and using the fact

in Remark 9 we arrive at (vi) Noting ℎ4149 = 1ℎ449 fromLemma 1(ii) we complete the proof of (vii)


In this section we find some new values of the parameter ℎ2119899by using the values of ℎ4119899 evaluated in Section 4 and in [6]

Theorem 11 One has

(i) ℎ26 = radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2

(ii) ℎ232 = (2 minus radic2 +

radic18 minus 12radic2)(2radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2)

(iii) ℎ16 = 1radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2

(iv) ℎ223 = 2radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2(2 minus

radic2 + radic18 minus 12radic2)

Proof Setting 119896 = 2 and 119899 = 3 in Lemma 2 we deduce that

ℎ43 = ℎ26ℎ232 (51)


From [6 page 174 Theorem 43(vii)] we have

ℎ43 =

(2 minus radic2 + radic18 minus 12radic2)

2

(52)

Combining (51) and (52) we obtain

ℎ26ℎ232 =


2

(53)

Next setting 119899 = 16 in Lemma 3 and simplifying usingLemma 1(ii) we obtain

radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)

Invoking (53) in (54) and simplifying we deduce that

(ℎ26

ℎ232

) =

2(8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2)

2 minus radic2 + radic18 minus 12radic2

(55)

Multiplying (53) and (55) and simplifying we complete theproof of (i) Dividing (53) by (55) and simplifying we arriveat (ii) (iii) and (iv) follow from (i) and (ii) respectively andLemma 1(ii)

The proofs ofTheorems 12ndash15 are identical to the proof ofTheorem 11 So we omit details and give only references of therequired results to prove them

Theorem 12 One has(i) ℎ210 = radic(4 + 1198882)radic2 minus 41198882

(ii) ℎ252 = 119888radic(4 + 1198882)radic2 minus 4119888

(iii) ℎ2110 = 2radic(4 + 1198882)radic2 minus 4119888

(iv) ℎ225 = radic(4 + 1198882)radic2 minus 4119888119888

where 119888 = radic22 + 10radic5 minus radic2 minus radic10 minus


2

minus 4

Proof of Theorem 12 follows from Theorem 10(i)Lemma 2 with 119896 = 2 and 119899 = 5 Lemma 1(ii) and Lemma 3with 119899 = 110

Theorem 13 One has(i) ℎ250 = radicminus2119889 + radic2(1198892 + 1)

(ii) ℎ2252 = 119889radicminus2119889 + radic2(1198892 + 1)

(iii) ℎ2150 = 1radicminus2119889 + radic2(1198892 + 1)

(iv) ℎ2225 = radicminus2119889 + radic2(1198892 + 1)119889

where 119889 = 114 minus 80radic2 + radic25965 minus 18360radic2 minus


2

minus 1

To prove Theorem 13 we use Theorem 10(iii) Lemma 2with 119896 = 2 and 119899 = 25 Lemma 1(ii) and Lemma 3 with 119899 =150

Theorem 14 One has

(i) ℎ214 = radic(minus8 + 6radic2)(12 minus 7radic2 minus radic238 minus 168radic2)

(ii) ℎ272 = radic12 minus 7radic2 minus radic238 minus 168radic22radicminus8 + 6radic2

(iii) ℎ2114 = 1radic(minus8 + 6radic2)(12 minus 7radic2 minus radic238 minus 168radic2)

(iv) ℎ227 = 2radicminus8 + 6radic2radic12 minus 7radic2 minus radic238 minus 168radic2

We employ Theorem 10(v) Lemma 2 with 119896 = 2 and119899 = 7 Lemma 1(ii) and Lemma 3 with 119899 = 114 to proveTheorem 14

Theorem 15 One has

(i) ℎ298 = radic2(119886 + radic119887 minus radic(119886 + radic119887)2minus 1)(radic2119886 + radic2119887 minus 1)

(ii) ℎ2492=radic(119886+radic119887 minusradic(119886+radic119887)2minus 1)2(radic2119886 + radic2119887 minus 1)

(iii) ℎ2198=1radic2(119886+radic119887minusradic(119886+radic119887)2minus 1)(radic2119886+radic2119887 minus 1)

(iv) ℎ2249=radic2(radic2119886+radic2119887 minus 1)(119886+radic119887 minus radic(119886+radic119887)2minus 1)

where a and b are given in Theorem 10(viii)

Proof follows fromTheorem 10(vii) Lemma 2 with 119896 = 2

and 119899 = 49 Lemma 1(ii) and Lemma 3 with 119899 = 198

6 Applications of ℎ4119899 and ℎ2119899

In this section we use the new values of the parameters ℎ4119899and ℎ2119899 to find explicit values of continued fractions 119888(119902) and119870(119902) defined in (3) and (4) respectively

The parameter ℎ4119899 is useful in finding explicit valuesof the continued fraction 119888(119902) If we know values of theparameter ℎ4119899 for any positive real number 119899 then explicitvalues of 119888(119890minus120587radic1198992) can be calculated by appealing to thefollowing theorem

Theorem 16 (see [6 page 177 Theorem 51]) One has

119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)

where 119899 is any positive real number


For example employing the value of ℎ45 fromTheorem 10(i) in Theorem 16 we obtain

119888 (119890minus120587radic52

)

= radic2 times (radic22 + 10radic5 minus radic2 minus radic10

minusradic(radic2 + radic10 minus radic22 + 10radic5)

2

minus 4)

minus1

(57)

Similarly we can find new values of 119888(119890minus1205872radic5) 119888(119890minus51205872)119888(119890minus12058710

) 119888(119890minus120587radic72) 119888(119890minus1205872radic7) 119888(119890minus71205872) and 119888(119890minus12058714

) byemploying the values of ℎ4119899 from Theorem 10(ii)ndash(viii)respectively in Theorem 16 Since it is a routine calculationwe omit details

Next the parameter ℎ2119899 is connected to continuedfraction119870(119902) by the following theorem

Theorem 17 (see [8 page 281Theorem 41]) For any positivereal number n one has

1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)

From Theorem 17 we note that if the values of ℎ2119899 areknown then the values of119870(119890minus120587radic1198992) can easily be evaluatedFor example using the value of ℎ26 from Theorem 11(i) inTheorem 17 we evaluate

119870(119890minus120587radic62

)

=

radicradicradicradic

radic

214radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2 minus 1

214radic8radic2 minus 10 + (radic2 minus 2)radic18 minus 12radic2 + 1

(59)

Similarly we can evaluate new values of 119870(119890minus120587radic1198992) for119899 =32 16 23 10 52 110 25 50 252 150 225 1472 114 27 98 492 and 249 by using Theorem 17 and thevalues of ℎ2119899 evaluated inTheorems 11ndash15

Acknowledgment

The author is thankful to the University Grants CommissionNew Delhi India for partially supporting the research workunder the Grant no F No 41-13942012(SR)

References

[1] E TWhittaker andG NWatsonACourse ofModern AnalysisCambridge Mathematical Library Cambridge University PressCambridge UK 1996 Indian edition is published by UniversalBook Stall New Delhi India 1991

[2] S Ramanujan Notebooks (2 Volumes) Tata Institute of Funda-mental Research Mumbai India 1957

[3] B C Berndt and H H Chan ldquoRamanujanrsquos explicit values forthe classical theta-functionrdquo Mathematika vol 42 no 2 pp278ndash294 1995

[4] B C Berndt Ramanujanrsquos Notebooks Part V Springer NewYork NY USA 1998

[5] J Yi ldquoTheta-function identities and the explicit formulas fortheta-function and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 381ndash400 2004

[6] N Saikia ldquoSome new identities for Ramanujanrsquos theta-function120601(q) and applicationsrdquo Far East Journal of Mathematical Sci-ences vol 44 no 2 pp 167ndash179 2010

[7] C Adiga and N Anitha ldquoA note on a continued fraction ofRamanujanrdquo Bulletin of the Australian Mathematical Societyvol 70 no 3 pp 489ndash497 2004

[8] ND Baruah andN Saikia ldquoExplicit evaluations of Ramanujan-Gollnitz-Gordon continued fractionrdquo Monatshefte fur Mathe-matik vol 154 no 4 pp 271ndash288 2008

[9] H H Chan and S-S Huang ldquoOn the Ramanujan-Gollnitz-Gordon continued fractionrdquoThe Ramanujan Journal vol 1 no1 pp 75ndash90 1997

[10] B C Berndt Ramanujanrsquos Notebooks Part III Springer NewYork NY USA 1991

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of the parameter ℎ4119899 In Section 5 we evaluate some newvalues of the parameter ℎ2119899 Finally in Section 6 we giveapplications of these values of ℎ4119899 and ℎ2119899 for the explicitevaluations of Ramanujanrsquos continued fractions 119888(119902) and119870(119902)

We end the introduction by defining Ramanujanrsquos mod-ular equation The complete elliptic integral of the first kind119870(119896) is defined by

119870 (119896) = int

1205872

0

119889120601

radic1 minus 1198962sin2120601=120587

2

infin

sum

119899=0

(12)2

119899

(119899)21198962119899

=120587

221198651 (

1

21

2 1 1198962)

(5)

where 0 lt 119896 lt 1 21198651 denotes the ordinary or Gaussianhypergeometric function and

(119886)119899 = 119886 (119886 + 1) (119886 + 2) sdot sdot sdot (119886 + 119899 minus 1) (6)

The number 119896 is called the modulus of 119870 and 1198961015840

=

radic1 minus 1198962 is called the complementary modulus Let 1198701198701015840 119871and 119871

1015840 denote the complete elliptic integrals of the firstkind associated with the moduli 119896 1198961015840 119897 and 1198971015840 respectivelySuppose that the equality

1198991198701015840

119870=1198711015840

119871(7)

holds for some positive integer 119899 Then a modular equationof degree 119899 is a relation between the moduli 119896 and 119897 which isimplied by (7)

If we set

119902 = exp(minus1205871198701015840

119870) 119902

1015840= exp(minus120587119871

1015840

119871) (8)

we see that (7) is equivalent to the relation 119902119899= 1199021015840 Thus

a modular equation can be viewed as an identity involvingtheta-functions at the arguments 119902 and 119902

119899 Ramanujanrecorded his modular equations in terms of 120572 and 120573 where120572 = 119896

2 and 120573 = 1198972 We say that 120573 has degree 119899 over 120572 The

multiplier 119898 connecting 120572 and 120573 is defined by 119898 = 1199111119911119899

where 119911119903 = 1206012(119902119903)

2 Preliminary Results

Lemma 1 (see [5 page 385 Theorem 22]) For positive realnumbers 119896 and 119899

(i) ℎ1198961 = 1(ii) ℎ1198961119899 = 1ℎ119896119899

Lemma 2 (see [5 page 387 Corollary 26]) For positive realnumbers 119896 and 119899

ℎ1198962 119899 = ℎ119896119899119896ℎ119896119899119896 (9)

Lemma 3 (see [5 page 392 Theorem 46]) One has

radic2(ℎ22119899ℎ24119899 +1

ℎ22119899ℎ24119899

) =ℎ24119899

ℎ2119899

+ 2 (10)

for any positive real number n

Lemma 4 (see [10 page 122 Entry 10 (i) and (v)]) One has

(i) 120601(119902) = radic119911

(ii) 120601(1199024) = radic119911(1 + (1 minus 120572)14))2

Lemma 5 (see [10 page 280-281 Entry 13(vii)]) If 120573 hasdegree 5 over 120572 then

(1205721205733)18

+ (1 minus 120572)(1 minus 120573)318

= (1205723120573)18

+ (1 minus 120572)3(1 minus 120573)

18

(11)

Lemma 6 (see [10 page 314 Entry 19(i)]) If 120573 has degree 7over 120572 then

(120572120573)18

+ (1 minus 120572)(1 minus 120573)18

= 1 (12)

3 Two New Identities for 120601(119902)

In this section we prove two new identities for theta-function 120601(119902) by using Ramanujanrsquos modular equations andtransformation formulas

Theorem 7 If 119875 = 120601(119902)120601(1199024) 119876 = 120601(119902

5)120601(11990220) then

1198756minus 256119875119876 + 640119875

2119876 minus 640119875

3119876

+ 3201198754119876 minus 70119875

5119876 + 640119875119876

2minus 1600119875

21198762

+ 160011987531198762minus 785119875

41198762+ 160119875

51198762minus 640119875119876

3

+ 160011987521198763minus 1620119875

31198763+ 800119875

41198763minus 160119875

51198763

+ 3201198751198764minus 785119875

21198764+ 800119875

31198764minus 400119875

41198764

+ 8011987551198764minus 70119875119876

5+ 160119875

21198765minus 160119875

31198765+ 80119875

41198765

minus 1611987551198765+ 1198766= 0

(13)

Proof Transcribing 119875 and 119876 using Lemma 4(i) and (iv) andthen simplifying we get

(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (14)


120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(15)



(1205721205733)18

minus (1205723120573)18

= (1 minus 120572)3(1 minus 120573)

18


(16)


(1205721205733)14

+ (1205723120573)14

= 119909 + 2(120572120573)12 (17)

where

119909 = (1 minus 120572)3(1 minus 120573)

14

+ (1 minus 120572) (1 minus 120573)314


(18)


(1205721205733)12

+ (1205723120573)12

= 119910 + 4119909(120572120573)12 (19)

where 119910 = 1199092+ 2120572120573


1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573 = 8119909119910(120572120573)

12 (20)


(1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573)2

minus 6411990921199102120572120573 = 0

(21)


119891 (119875119876) 119892 (119875 119876) ℎ (119875 119876) = 0 (22)

where

119891 (119875119876) = (119875 minus 119876)4

119892 (119875 119876) = 1198756minus 256119875119876 + 640119875

2119876

minus 6401198753119876 + 320119875

4119876 minus 70119875

5119876 + 640119875119876

2

minus 160011987521198762+ 1600119875

31198762minus 785119875

41198762+ 160119875

51198762

minus 6401198751198763+ 1600119875

21198763minus 1620119875

31198763+ 800119875

41198763

minus 16011987551198763+ 320119875119876

4minus 785119875

21198764+ 800119875

31198764

minus 40011987541198764+ 80119875

51198764minus 70119875119876

5+ 160119875

21198765

minus 16011987531198765+ 80119875

41198765minus 16119875

51198765+ 1198766

ℎ (119875 119876)

= 1611987510minus 96119875

9119876 minus 32119875

10119876 + 240119875

81198762+ 192119875

91198762

+ 24119875101198762minus 320119875

71198763minus 480119875

81198763minus 144119875

91198763

minus 8119875101198763+ 256119875

61198764+ 608119875

71198764+ 384119875

81198764

+ 4011987591198764+ 119875101198764minus 448119875

51198765+ 352119875

61198765

minus 126411987571198765+ 248119875

81198765minus 70119875

91198765+ 256119875

41198766

+ 35211987551198766minus 880119875

61198766+ 1640119875

71198766minus 785119875

81198766

+ 16011987591198766minus 320119875

31198767+ 608119875

41198767minus 1264119875

51198767

+ 164011987561198767minus 1620119875

71198767+ 800119875

81198767minus 160119875

91198767

+ 24011987521198768minus 480119875

31198768+ 384119875

41198768+ 248119875

51198768

minus 78511987561198768+ 800119875

71198768minus 400119875

81198768+ 80119875

91198768

minus 961198751198769+ 192119875

21198769minus 144119875

31198769+ 40119875

41198769

minus 7011987551198769+ 160119875

61198769minus 160119875

71198769+ 80119875

81198769

minus 1611987591198769+ 16119876

10minus 32119875119876

10+ 24119875

211987610

minus 8119875311987610+ 119875411987610

(23)



Theorem 8 If 119875 = 120601(119902)120601(1199024) 119876 = 120601(119902

7)120601(11990228) then

1198758minus 4096119875119876 + 14336119875

2119876 minus 21504119875

3119876 + 17920119875

4119876

minus 87361198755119876 + 2352119875

6119876 minus 280119875

7119876 + 14336119875119876

2

minus 5196811987521198762+ 80640119875

31198762minus 69440119875

41198762+ 35056119875

51198762

minus 977211987561198762+ 1176119875

71198762minus 21504119875119876

3+ 80640119875

21198763

minus 12947211987531198763+ 115360119875

41198763minus 60424119875

51198763+ 17528119875

61198763

minus 218411987571198763+ 17920119875119876

4minus 69440119875

21198764+ 115360119875

31198764

minus 10633011987541198764+ 57680119875

51198764minus 17360119875

61198764+ 2240119875

71198764

minus 87361198751198765+ 35056119875

21198765minus 60424119875

31198765+ 57680119875

41198765

minus 3236811987551198765+ 10080119875

61198765minus 1344119875

71198765+ 2352119875119876

6

minus 977211987521198766+ 17528119875

31198766minus 17360119875

41198766+ 10080119875

51198766

minus 324811987561198766+ 448119875

71198766minus 280119875119876

7+ 1176119875

21198767

minus 218411987531198767+ 2240119875

41198767minus 1344119875

51198767+ 448119875

61198767

minus 6411987571198767+ 1198768= 0

(24)



(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (25)


120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(26)


(1 minus 120572) (1 minus 120573)18

= 1 minus (120572120573)18 (27)


119909 minus (120572120573)14

= minus2(120572120573)18 (28)



1199092+ (120572120573)

12= (4 + 2119909) (120572120573)

14 (29)


1199094+ 120572120573 = ((4 + 2119909)

2minus 21199092) (120572120573)

12 (30)


(1199094+ 120572120573)

2

= ((4 + 2119909)2minus 21199092)2

(120572120573) (31)





Theorem 10 One has



2

minus 4)2



2

minus 4)2



2

minus 1



2

minus 1






and 119887 = 6940535 minus 4907700radic2 +




119875 = radic2ℎ4119899 119876 = radic2ℎ425119899 (32)


(ℎ6

45+ ℎminus6

45) minus 70 (ℎ

4

45+ ℎminus4

45) + 320radic2 (ℎ

3

45+ ℎminus3

45)

minus 1425 (ℎ2

45+ ℎminus2

45) + 1920radic2 (ℎ45 + ℎ

minus1

45) minus 3348 = 0

(33)

Equivalently

1198606minus 76119860

4+ 320radic2119860

3minus 1136119860

2+ 960radic2119860 minus 640 = 0

(34)

where

119860 = ℎ45 + ℎminus1

45 (35)





(ℎ2

425+ ℎminus2


minus1

425)


(37)

Equivalently



where

119861 = ℎ425 + ℎminus1

425 (39)





Theorem 8 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ449119899 (41)


ℎ16

47minus 280ℎ

14

47+ 2352radic2ℎ

13

47minus 18508ℎ

12

47

+ 44016radic2ℎ11

47minus 140616ℎ

10

47+ 159264radic2ℎ

9

47

minus 262810ℎ8

47+ 159264radic2ℎ

7

47minus 140616ℎ

6

47

+ 44016radic2ℎ5

47minus 18508ℎ

4

47+ 2352radic2ℎ

3

47

minus 280ℎ2

47+ 1 = 0

(42)



(ℎ8

47+ ℎminus8

47) minus 280 (ℎ

6

47+ ℎminus6

47) + 2352radic2 (ℎ

5

47+ ℎminus5

47)

minus 18508 (ℎ4

47+ ℎminus4

47) + 44016radic2 (ℎ

3

47+ ℎminus3

47)

minus 140616 (ℎ2

47+ ℎminus2

47) + 159264radic2 (ℎ47 + ℎ

minus1

47)

minus 262810 = 0

(43)

Equivalently

1198718minus 288119871

6+ 2352radic2119871

5minus 16808119871

4+ 32256radic2119871

3minus 69120119871

2

+ 38976radic2119871 minus 18032 = 0

(44)

where

119871 = ℎ47 + ℎminus1

47 (45)


119871 = 12 minus 7radic2 (46)



ℎ4

449+ ℎminus4

449+ 4 (minus1317 + 931radic2) (ℎ

3

449+ ℎminus3

449)

+ 28 (minus2053 + 1452radic2) (ℎ2

449+ ℎminus2

449)


449)

+ 210 (minus837 + 592radic2) = 0

(47)

Equivalently

1198634+ 8 (minus1317 + 931radic2)119863

3+ 16 (minus3593 + 2541radic2)119863

2


(48)

where

119863 = ℎ449 + ℎminus1

449 (49)


119863 = 2 (119886 + radic119887) (50)


and 119887 = 6940535 minus 4907700radic2 +






Theorem 11 One has








ℎ43 = ℎ26ℎ232 (51)



ℎ43 =


2

(52)


ℎ26ℎ232 =


2

(53)


radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)


(ℎ26

ℎ232

) =



(55)









2

minus 4








2

minus 1


Theorem 14 One has






Theorem 15 One has












119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)




119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1205721205733)18

minus (1205723120573)18

= (1 minus 120572)3(1 minus 120573)

18


(16)


(1205721205733)14

+ (1205723120573)14

= 119909 + 2(120572120573)12 (17)

where

119909 = (1 minus 120572)3(1 minus 120573)

14

+ (1 minus 120572) (1 minus 120573)314


(18)


(1205721205733)12

+ (1205723120573)12

= 119910 + 4119909(120572120573)12 (19)

where 119910 = 1199092+ 2120572120573


1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573 = 8119909119910(120572120573)

12 (20)


(1205721205733+ 1205723120573 + 2120572

21205732minus 1199102minus 16119909

2120572120573)2

minus 6411990921199102120572120573 = 0

(21)


119891 (119875119876) 119892 (119875 119876) ℎ (119875 119876) = 0 (22)

where

119891 (119875119876) = (119875 minus 119876)4

119892 (119875 119876) = 1198756minus 256119875119876 + 640119875

2119876

minus 6401198753119876 + 320119875

4119876 minus 70119875

5119876 + 640119875119876

2

minus 160011987521198762+ 1600119875

31198762minus 785119875

41198762+ 160119875

51198762

minus 6401198751198763+ 1600119875

21198763minus 1620119875

31198763+ 800119875

41198763

minus 16011987551198763+ 320119875119876

4minus 785119875

21198764+ 800119875

31198764

minus 40011987541198764+ 80119875

51198764minus 70119875119876

5+ 160119875

21198765

minus 16011987531198765+ 80119875

41198765minus 16119875

51198765+ 1198766

ℎ (119875 119876)

= 1611987510minus 96119875

9119876 minus 32119875

10119876 + 240119875

81198762+ 192119875

91198762

+ 24119875101198762minus 320119875

71198763minus 480119875

81198763minus 144119875

91198763

minus 8119875101198763+ 256119875

61198764+ 608119875

71198764+ 384119875

81198764

+ 4011987591198764+ 119875101198764minus 448119875

51198765+ 352119875

61198765

minus 126411987571198765+ 248119875

81198765minus 70119875

91198765+ 256119875

41198766

+ 35211987551198766minus 880119875

61198766+ 1640119875

71198766minus 785119875

81198766

+ 16011987591198766minus 320119875

31198767+ 608119875

41198767minus 1264119875

51198767

+ 164011987561198767minus 1620119875

71198767+ 800119875

81198767minus 160119875

91198767

+ 24011987521198768minus 480119875

31198768+ 384119875

41198768+ 248119875

51198768

minus 78511987561198768+ 800119875

71198768minus 400119875

81198768+ 80119875

91198768

minus 961198751198769+ 192119875

21198769minus 144119875

31198769+ 40119875

41198769

minus 7011987551198769+ 160119875

61198769minus 160119875

71198769+ 80119875

81198769

minus 1611987591198769+ 16119876

10minus 32119875119876

10+ 24119875

211987610

minus 8119875311987610+ 119875411987610

(23)



Theorem 8 If 119875 = 120601(119902)120601(1199024) 119876 = 120601(119902

7)120601(11990228) then

1198758minus 4096119875119876 + 14336119875

2119876 minus 21504119875

3119876 + 17920119875

4119876

minus 87361198755119876 + 2352119875

6119876 minus 280119875

7119876 + 14336119875119876

2

minus 5196811987521198762+ 80640119875

31198762minus 69440119875

41198762+ 35056119875

51198762

minus 977211987561198762+ 1176119875

71198762minus 21504119875119876

3+ 80640119875

21198763

minus 12947211987531198763+ 115360119875

41198763minus 60424119875

51198763+ 17528119875

61198763

minus 218411987571198763+ 17920119875119876

4minus 69440119875

21198764+ 115360119875

31198764

minus 10633011987541198764+ 57680119875

51198764minus 17360119875

61198764+ 2240119875

71198764

minus 87361198751198765+ 35056119875

21198765minus 60424119875

31198765+ 57680119875

41198765

minus 3236811987551198765+ 10080119875

61198765minus 1344119875

71198765+ 2352119875119876

6

minus 977211987521198766+ 17528119875

31198766minus 17360119875

41198766+ 10080119875

51198766

minus 324811987561198766+ 448119875

71198766minus 280119875119876

7+ 1176119875

21198767

minus 218411987531198767+ 2240119875

41198767minus 1344119875

51198767+ 448119875

61198767

minus 6411987571198767+ 1198768= 0

(24)



(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (25)


120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(26)


(1 minus 120572) (1 minus 120573)18

= 1 minus (120572120573)18 (27)


119909 minus (120572120573)14

= minus2(120572120573)18 (28)



1199092+ (120572120573)

12= (4 + 2119909) (120572120573)

14 (29)


1199094+ 120572120573 = ((4 + 2119909)

2minus 21199092) (120572120573)

12 (30)


(1199094+ 120572120573)

2

= ((4 + 2119909)2minus 21199092)2

(120572120573) (31)





Theorem 10 One has



2

minus 4)2



2

minus 4)2



2

minus 1



2

minus 1






and 119887 = 6940535 minus 4907700radic2 +




119875 = radic2ℎ4119899 119876 = radic2ℎ425119899 (32)


(ℎ6

45+ ℎminus6

45) minus 70 (ℎ

4

45+ ℎminus4

45) + 320radic2 (ℎ

3

45+ ℎminus3

45)

minus 1425 (ℎ2

45+ ℎminus2

45) + 1920radic2 (ℎ45 + ℎ

minus1

45) minus 3348 = 0

(33)

Equivalently

1198606minus 76119860

4+ 320radic2119860

3minus 1136119860

2+ 960radic2119860 minus 640 = 0

(34)

where

119860 = ℎ45 + ℎminus1

45 (35)





(ℎ2

425+ ℎminus2


minus1

425)


(37)

Equivalently



where

119861 = ℎ425 + ℎminus1

425 (39)





Theorem 8 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ449119899 (41)


ℎ16

47minus 280ℎ

14

47+ 2352radic2ℎ

13

47minus 18508ℎ

12

47

+ 44016radic2ℎ11

47minus 140616ℎ

10

47+ 159264radic2ℎ

9

47

minus 262810ℎ8

47+ 159264radic2ℎ

7

47minus 140616ℎ

6

47

+ 44016radic2ℎ5

47minus 18508ℎ

4

47+ 2352radic2ℎ

3

47

minus 280ℎ2

47+ 1 = 0

(42)



(ℎ8

47+ ℎminus8

47) minus 280 (ℎ

6

47+ ℎminus6

47) + 2352radic2 (ℎ

5

47+ ℎminus5

47)

minus 18508 (ℎ4

47+ ℎminus4

47) + 44016radic2 (ℎ

3

47+ ℎminus3

47)

minus 140616 (ℎ2

47+ ℎminus2

47) + 159264radic2 (ℎ47 + ℎ

minus1

47)

minus 262810 = 0

(43)

Equivalently

1198718minus 288119871

6+ 2352radic2119871

5minus 16808119871

4+ 32256radic2119871

3minus 69120119871

2

+ 38976radic2119871 minus 18032 = 0

(44)

where

119871 = ℎ47 + ℎminus1

47 (45)


119871 = 12 minus 7radic2 (46)



ℎ4

449+ ℎminus4

449+ 4 (minus1317 + 931radic2) (ℎ

3

449+ ℎminus3

449)

+ 28 (minus2053 + 1452radic2) (ℎ2

449+ ℎminus2

449)


449)

+ 210 (minus837 + 592radic2) = 0

(47)

Equivalently

1198634+ 8 (minus1317 + 931radic2)119863

3+ 16 (minus3593 + 2541radic2)119863

2


(48)

where

119863 = ℎ449 + ℎminus1

449 (49)


119863 = 2 (119886 + radic119887) (50)


and 119887 = 6940535 minus 4907700radic2 +






Theorem 11 One has








ℎ43 = ℎ26ℎ232 (51)



ℎ43 =


2

(52)


ℎ26ℎ232 =


2

(53)


radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)


(ℎ26

ℎ232

) =



(55)









2

minus 4








2

minus 1


Theorem 14 One has






Theorem 15 One has












119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)




119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1 minus 120572)14

=2

119875minus 1 (1 minus 120573)

14= (

2

119876minus 1) (25)


120572 = 1 minus (2

119875minus 1)

4

120573 = 1 minus (2

119876minus 1)

4

(26)


(1 minus 120572) (1 minus 120573)18

= 1 minus (120572120573)18 (27)


119909 minus (120572120573)14

= minus2(120572120573)18 (28)



1199092+ (120572120573)

12= (4 + 2119909) (120572120573)

14 (29)


1199094+ 120572120573 = ((4 + 2119909)

2minus 21199092) (120572120573)

12 (30)


(1199094+ 120572120573)

2

= ((4 + 2119909)2minus 21199092)2

(120572120573) (31)





Theorem 10 One has



2

minus 4)2



2

minus 4)2



2

minus 1



2

minus 1






and 119887 = 6940535 minus 4907700radic2 +




119875 = radic2ℎ4119899 119876 = radic2ℎ425119899 (32)


(ℎ6

45+ ℎminus6

45) minus 70 (ℎ

4

45+ ℎminus4

45) + 320radic2 (ℎ

3

45+ ℎminus3

45)

minus 1425 (ℎ2

45+ ℎminus2

45) + 1920radic2 (ℎ45 + ℎ

minus1

45) minus 3348 = 0

(33)

Equivalently

1198606minus 76119860

4+ 320radic2119860

3minus 1136119860

2+ 960radic2119860 minus 640 = 0

(34)

where

119860 = ℎ45 + ℎminus1

45 (35)





(ℎ2

425+ ℎminus2


minus1

425)


(37)

Equivalently



where

119861 = ℎ425 + ℎminus1

425 (39)





Theorem 8 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ449119899 (41)


ℎ16

47minus 280ℎ

14

47+ 2352radic2ℎ

13

47minus 18508ℎ

12

47

+ 44016radic2ℎ11

47minus 140616ℎ

10

47+ 159264radic2ℎ

9

47

minus 262810ℎ8

47+ 159264radic2ℎ

7

47minus 140616ℎ

6

47

+ 44016radic2ℎ5

47minus 18508ℎ

4

47+ 2352radic2ℎ

3

47

minus 280ℎ2

47+ 1 = 0

(42)



(ℎ8

47+ ℎminus8

47) minus 280 (ℎ

6

47+ ℎminus6

47) + 2352radic2 (ℎ

5

47+ ℎminus5

47)

minus 18508 (ℎ4

47+ ℎminus4

47) + 44016radic2 (ℎ

3

47+ ℎminus3

47)

minus 140616 (ℎ2

47+ ℎminus2

47) + 159264radic2 (ℎ47 + ℎ

minus1

47)

minus 262810 = 0

(43)

Equivalently

1198718minus 288119871

6+ 2352radic2119871

5minus 16808119871

4+ 32256radic2119871

3minus 69120119871

2

+ 38976radic2119871 minus 18032 = 0

(44)

where

119871 = ℎ47 + ℎminus1

47 (45)


119871 = 12 minus 7radic2 (46)



ℎ4

449+ ℎminus4

449+ 4 (minus1317 + 931radic2) (ℎ

3

449+ ℎminus3

449)

+ 28 (minus2053 + 1452radic2) (ℎ2

449+ ℎminus2

449)


449)

+ 210 (minus837 + 592radic2) = 0

(47)

Equivalently

1198634+ 8 (minus1317 + 931radic2)119863

3+ 16 (minus3593 + 2541radic2)119863

2


(48)

where

119863 = ℎ449 + ℎminus1

449 (49)


119863 = 2 (119886 + radic119887) (50)


and 119887 = 6940535 minus 4907700radic2 +






Theorem 11 One has








ℎ43 = ℎ26ℎ232 (51)



ℎ43 =


2

(52)


ℎ26ℎ232 =


2

(53)


radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)


(ℎ26

ℎ232

) =



(55)









2

minus 4








2

minus 1


Theorem 14 One has






Theorem 15 One has












119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)




119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119861 = ℎ425 + ℎminus1

425 (39)





Theorem 8 we get

119875 = radic2ℎ4119899 119876 = radic2ℎ449119899 (41)


ℎ16

47minus 280ℎ

14

47+ 2352radic2ℎ

13

47minus 18508ℎ

12

47

+ 44016radic2ℎ11

47minus 140616ℎ

10

47+ 159264radic2ℎ

9

47

minus 262810ℎ8

47+ 159264radic2ℎ

7

47minus 140616ℎ

6

47

+ 44016radic2ℎ5

47minus 18508ℎ

4

47+ 2352radic2ℎ

3

47

minus 280ℎ2

47+ 1 = 0

(42)



(ℎ8

47+ ℎminus8

47) minus 280 (ℎ

6

47+ ℎminus6

47) + 2352radic2 (ℎ

5

47+ ℎminus5

47)

minus 18508 (ℎ4

47+ ℎminus4

47) + 44016radic2 (ℎ

3

47+ ℎminus3

47)

minus 140616 (ℎ2

47+ ℎminus2

47) + 159264radic2 (ℎ47 + ℎ

minus1

47)

minus 262810 = 0

(43)

Equivalently

1198718minus 288119871

6+ 2352radic2119871

5minus 16808119871

4+ 32256radic2119871

3minus 69120119871

2

+ 38976radic2119871 minus 18032 = 0

(44)

where

119871 = ℎ47 + ℎminus1

47 (45)


119871 = 12 minus 7radic2 (46)



ℎ4

449+ ℎminus4

449+ 4 (minus1317 + 931radic2) (ℎ

3

449+ ℎminus3

449)

+ 28 (minus2053 + 1452radic2) (ℎ2

449+ ℎminus2

449)


449)

+ 210 (minus837 + 592radic2) = 0

(47)

Equivalently

1198634+ 8 (minus1317 + 931radic2)119863

3+ 16 (minus3593 + 2541radic2)119863

2


(48)

where

119863 = ℎ449 + ℎminus1

449 (49)


119863 = 2 (119886 + radic119887) (50)


and 119887 = 6940535 minus 4907700radic2 +






Theorem 11 One has








ℎ43 = ℎ26ℎ232 (51)



ℎ43 =


2

(52)


ℎ26ℎ232 =


2

(53)


radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)


(ℎ26

ℎ232

) =



(55)









2

minus 4








2

minus 1


Theorem 14 One has






Theorem 15 One has












119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)




119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ43 =


2

(52)


ℎ26ℎ232 =


2

(53)


radic2 (ℎ26ℎ232 + (ℎ26ℎ232)minus1) = (

ℎ26

ℎ232

) + 2 (54)


(ℎ26

ℎ232

) =



(55)









2

minus 4








2

minus 1


Theorem 14 One has






Theorem 15 One has












119888 (119890minus120587radic1198992

) =1

radic2ℎ4119899

(56)




119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888 (119890minus120587radic52

)



2

minus 4)

minus1

(57)






1198702(119890minus120587radic1198992

) =214ℎ2119899 minus 1

214ℎ2119899 + 1 (58)


119870(119890minus120587radic62

)

=


radic



(59)


Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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