Journal of Statistical Planning andInference 135 (2005) 18–39

www.elsevier.com/locate/jspi

On a class of polynomials related to classicalorthogonal and Fibonacci polynomials with

probability applications

Demetrios L. Antzoulakos, Markos V. Koutras∗Department of Statistics and Insurance Science, University of Piraeus, 80, Karaoli & Dimitriou Str, Piraeus

18534, Greece

Received 26 August 2002; received in revised form 4 May 2003; accepted 7 February 2005Available online 22 March 2005

Abstract

In the present work, a class of polynomials is introduced which is defined by a certain recursivescheme associating then-th member of the class to the previousk�1 members. The class is wideenough to accommodate most of the well-known families of orthogonal polynomials (Legendre,Hermite, Chebyshev, Laguerre, etc.) as well as the recently studied Fibonacci polynomials of orderkand several extensions of them.Using a generating function approach, it is proved that the new class is closed under the operations

of convolution and partial summation. Finally the general results developed here are exploited in theinvestigation of a number of probability applications pertaining to the occurrence of runs and scansin sequences of binary trials.© 2005 Elsevier B.V. All rights reserved.

MSC:60C05; 62E15; 11B39

Keywords:Orthogonal polynomials; Fibonacci polynomials; Convolution; Partial sum; Waiting time; Successruns; Probability mass function; Cumulative distribution function

∗ Corresponding author. Tel.: +3014142393; fax: +3014142340.E-mail address:mkoutras@unipi.gr(M.V. Koutras).

0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jspi.2005.02.003

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3919

1. Introduction

The classical sequences of (orthogonal) polynomials associated with the names of Her-mite, Laguerre, Chebyshev, Legendre and Gegenbauer are unquestionably the most ex-tensively studied and widely applied “special functions”. Because of their widespread ap-plicability in diverse scientific areas such as mathematical physics, approximation theory,quantum mechanics, etc., they became the subject of an enormous literature going backas far as the beginning of the previous century. The most thorough single accounts of theclassical polynomials may be found in the treatises bySansone (1959)andSzegö (1975).The classical orthogonal polynomialspn(x), n = 0,1, . . . satisfy three-term recurrence

relations of the form

pn+1(x)= (Anx + Bn)pn(x)− Cnpn−1(x), n�0

with initial conditionsp−1(x)=0,p0(x)=1. For anumber of alternative characterizationsofthem, mainly through differential equations the interested reader may consult any standardbook on special functions, e.g.Chichara (1978)or Lebedev (1965).In the present article we consider a class of polynomials which is defined by a certain

recursive scheme that is wide enough to accommodate almost all classical families oforthogonal polynomials, and at the same time some recently studied families of polynomialsassociated with applied probability models. Our scheme pertains a(k+ 1)-term recurrencerelation (k�1 is a given integer) with appropriately chosen coefficients so that the resultingclass holds closure properties for the operations of convolution and partial summation.The working environment provided by the new family of polynomials provides efficienttools for establishing a unified treatment of pattern occurrences in sequences of binarytrials.The organization of the present paper is as follows. In Section 2, we introduce the new

classof polynomials and illustrate how the classical orthogonal polynomials and the recentlystudied Fibonacci polynomials of orderk can be viewed as members of the new family.In Section 3, we establish several characterizations for the class in terms of generatingfunctions while Section 4 deals with a number of closure properties. Finally, Section 5exhibits how the general theory developed in the previous sections can be exploited in thederivation of results pertaining to pattern enumeration problems in sequences of binarytrials.

2. A new class of polynomials

Let k�1 be a fixed integer andai(x), bi(x), i=1,2, . . . , k be polynomials inxof degreeat most 1 such that at least one ofak(x) andbk(x) is different from zero. The class ofpolynomials defined by the recursive scheme

pn(x)=k∑i=1

(ai(x)+ bi(x)

n

)pn−i (x), n= 1,2, . . . (2.1)

20 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

will be denoted byPk(a,b) or, when no confusion is likely to arise, simply byPk. Through-out the remainder of this article, the convention will be maintained that, when a recurrenceformula of the previous form is written, the initial conditions are

pn(x)={0, n<0,1, n= 0

unless it is explicitly indicated otherwise.By way of example we mention that, the Legendre polynomials, defined by Rodrigues’

formula

pn(x)= 1

2nn!dn

dxn(x2− 1)n, n= 0,1, . . . (2.2)

are members of the classP2 with

a1(x)= 2x, a2(x)= −1, b1(x)= −x, b2(x)= 1 (2.3)

as it is easily verified if we recall the well-known recurrence

npn(x)= x(2n− 1)pn−1(x)− (n− 1)pn−2(x), n�1

satisfied by them.Another interesting special case ofPk results by considering the choice

ai(x)= x, bi(x)= 0 (2.4)

for all i = 1,2, . . . , k. The associated family of polynomials{pn(x)}n�0 obeys the recur-rence

pn(x)= xk∑i=1

pn−i (x), n�1 (2.5)

and it can be easily verified that the shifted sequence of polynomials

F (k)n (x)= pn−1(x), n= 1,2, . . . (2.6)

gives rise to the Fibonacci (type) polynomials of orderk which were recently studied byPhilippou et al. (1985), Philippou and Georghiou (1989)andPhilippou and Antzoulakos(1990).Table 1illustrates how the four classical families of orthogonal polynomials (Legendre,

Hermite, Laguerre and Chebyshev) can be viewed as special cases ofPk. Note that theChebyshev polynomials of the first kind do not fit exactly in the model presented earlier inthe sense thata2(x), b2(x) are polynomials of degree 2. However, this violation does notcause any major problem, since most of the unifying properties of the classPk that will bepresented later, remain valid under the less restrictive condition thatai(x), bi(x) are anypolynomials inx.

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3921

Table 1

Orthogonal polynomials Recurrence pn(x) k Coefficients:ai (x), bi (x)

LegendrePn(x)pn(x)= x(2− 1

n )pn−1(x)−(1− 1

n )pn−2(x)Pn(x) 2

a1 = 2xa2 = −1

b1 = −xb2 = 1

HermiteHn(x)pn(x)= 2x

n pn−1(x)− 2n pn−2(x)

Hn(x)n! 2

a1 = 0a2 = 0

b1 = 2xb2 = −2

LaguerreLan(x)pn(x)= (2+ a−x−1

n )pn−1(x)−(1+ a−1

n )pn−2(x)Lan(x) 2

a1 = 2a2 = −1

b1 = a − x − 1b2 = 1− a

LaguerreLan(x)pn(x)= (−1+ a−x+1

n )pn−1(x)− xnpn−2(x)

La−nn (x) 2a1 = −1a2 = 0

b1 = a − x + 1b2 = −x

ChebyshevTn(x)(first kind)

pn(x)= x(3− 2n )pn−1(x)

−(2x2 + 1− 4x2n )pn−2(x)

+x(1− 2n )pn−3(x)

Tn(x) 3a1 = 3xa2 = −(2x2 + 1)a3 = x

b1 = −2xb2 = 4x2b3 = −2x

ChebyshevUn(x)(second kind)

pn(x)= 2xpn−1(x)− pn−2(x) Un(x) 2a1 = 2xa2 = −1

b1 = 0b2 = 0

3. Generating functions and characterizations

A commonly used practice in the theory of orthogonal polynomials is to compute firsttheir generating function and subsequently exploit it to derive (usually in a very simple way)several interesting properties, e.g. integral expressions, difference-differential equations,asymptotic results, etc. In the present section we are going to establish several formulae forthe generating function of the familyPk, namely

p(x; t)=∞∑n=0

pn(x)tn. (3.1)

In the last expression, we assume that|t | is sufficiently small so that the power series inthe RHS converges. In the sequel, we shall be suppressingx in the notationsai(x), bi(x),i.e. we shall writeai , bi instead. Moreover, we shall make frequent use of the notations

A(x; t)=k∑i=1

ai(x)ti =

k∑i=1

aiti , B(x; t)=

k∑i=1

bi(x)ti =

k∑i=1

biti

(A(x; t), B(x; t) are polynomials inx of degree at most 1, and polynomials int of degreeat mostk with at least one of them being a polynomial int of degreek). The quantityp(x; t) regarded as a function oft, will be assumed to be analytic in the disk|t |� t0 (fort0 a sufficiently small positive number), a fact justifying the term by term power seriesdifferentiation used in the sequel.We are now ready to state and prove a number of interesting results for the classPk.

22 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Theorem 3.1. If {pn(x)}n�0 belongs to the classPk(a,b) then the generating function(3.1)satisfies the partial differential equation

�p(x; t)�t

= p(x; t)∑ki=1 (iai + bi)t i−11−∑k

i=1 ait i. (3.2)

Proof. Differentiating the power series in the RHS of (3.1) term by term, we get

�p(x; t)�t

=∞∑n=1

npn(x)tn−1

and substitutingpn(x) by virtue of the recursive scheme (2.1) we deduce

�p(x; t)�t

=∞∑n=1

(k∑i=1(nai + bi)pn−i (x)

)tn−1

=k∑i=1

∞∑n=i(nai + bi)pn−i (x)tn−1

=k∑i=1

∞∑n=0

[(n+ i)ai + bi]pn(x)tn+i−1

=k∑i=1

aiti

( ∞∑n=0

npn(x)tn−1

)+

k∑i=1(iai + bi)t i−1

( ∞∑n=0

pn(x)tn

).

This leads to the identity

�p(x; t)�t

=(

k∑i=1

aiti

)�p(x; t)

�t+(

k∑i=1(iai + bi)t i−1

)p(x; t)

from which the proof of the theorem follows immediately.�

It is evident that, following the reverse route,wemayeasily verify that the partial differen-tial equation (3.2) characterizes the familyPk. Therefore, if we have at hand the generatingfunctionp(x; t) of a sequence of polynomials{pn(x)}n�0 and after differentiating it withrespect tot we succeed in establishing an identity of the form (3.2), we can immediatelywrite down a recursive scheme of the form (2.1) for the polynomialspn(x), n= 0,1, . . . .The next corollary is a restatement of Theorem 3.1, where the functionsA(x; t), B(x; t)

are used instead of the (finite) sequencesai(x), bi(x), i = 1,2, . . . , k.

Corollary 3.1. If {pn(x)}n�0 belongs to the classPk(a,b) then the generating function(3.1)satisfies the partial differential equation

�p(x; t)�t

= p(x; t)�A(x; t)

�t+ 1tB(x; t)

1− A(x; t) .

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3923

Proof. It suffices to observe that

k∑i=1(iai + bi)t i−1=

k∑i=1

iai ti−1+ 1

t

k∑i=1

biti = �A(x; t)

�t+ 1tB(x; t). �

Yet another necessary and sufficient condition for a sequence of polynomials to belongto the familyPk is provided by the next corollary. This formulation seems to be preferableto the ones described in Theorem 3.1 or Corollary 3.1 both on grounds of simplicity andfunctionality.

Corollary 3.2. If {pn(x)}n�0 belongs to the classPk(a,b) then the generating function(3.1)satisfies the partial differential equation

�p(x; t)�t

= p(x; t) C(x; t)t[1− A(x; t)] , (3.3)

whereA(x; t), C(x; t) are polynomials in t of degree at most k(with at least one of thembeing a polynomial in t of degree k), and their coefficients are polynomials in x of degreeat most1.Conversely, if the generating function of a sequence of polynomials{pn(x)}n�0satisfies the partial differential equation(3.3)with

A(x; t)=k∑i=1

ai(x)ti , C(x; t)=

k∑i=1

ci(x)ti ,

whereai(x), ci(x), i = 1,2, . . . , k are polynomials in x of degree at most1, thenpn(x),n= 0,1, . . . satisfy the recurrence relation(2.1)with

bi(x)= ci(x)− iai(x), i = 1,2, . . . , k.

Proof. Results immediately from Corollary 3.1 on setting

C(x; t)= t �A(x; t)�t

+ B(x; t). �

As an illustration, let us consider the generating function

p(x; t)= (1+ t)a exp(−xt).Differentiating with respect tot we deduce

�p(x; t)�t

= p(x; t) a − x − xt1+ t

which can be written in form (3.3) with

A(x; t)= −t, C(x; t)= (a − x)t − xt2.

24 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Hence,k = 2,

ai(x)={−1, i = 1,0, i = 2, ci(x)=

{a − x, i = 1,−x, i = 2,

bi(x)={a − x + 1, i = 1,−x, i = 2

andp(x; t) is associatedwith a sequenceof polynomials{pn(x)}n�0 obeying the recurrence

pn(x)=(

−1+ a − x + 1n

)pn−1(x)+

(0− x

n

)pn−2(x), n�1.

Note thatpn(x)= La−nn (x) whereL�n(x) are the classical Laguerre polynomials; this is

easily ascertainableby comparing theaforementioned recursive relation to theonecontainedin the fourth row ofTable 1.As a final remark we mention that, ifA(x; t) andC(x; t) are available, one can compute

the generating functionp(x; t) by the aid of the formula

p(x; t)= exp{∫ t

0

C(x; s)s[1− A(x; s)] ds

}(3.4)

which results immediately by integrating the obvious by-product of (3.3)

��s

[lnp(x; s)] = C(x; s)s[1− A(x; s)] (3.5)

over the ranges ∈ [0, t] and taking into account thatp(x;0)= p0(x)= 1.

4. Properties of the classPk

In this section we are studying the algebra of the classPk. More specifically, we shallprove that the classPk is closed under the following two operations defined over the spaceof sequences of polynomials:

a. convolutionb. partial summation.

To prove the first assertion, let{p′n(x)}n�0 and{p′′

n(x)}n�0 be two sequences of poly-nomials belonging to the classesPk(a,b′) andPk(a,b′′) respectively. Then, it is clear thatthe corresponding generating functionsp′(x; t) andp′′(x; t) obey the partial differentialequations (see (3.5))

��t

[lnp′(x; t)] =�A(x; t)

�t+ 1tB ′(x; t)

1− A(x; t) ,

��t

[lnp′′(x; t)] =�A(x; t)

�t+ 1tB ′′(x; t)

1− A(x; t)

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3925

and on introducing the notation

D(x; t)= B ′(x; t)+ B ′′(x; t)+ t �A(x; t)�t

=k∑i=1(b′i + b′′

i + iai)t i

we may also write

��t

[ln(p′(x; t)p′′(x; t))] =�A(x; t)

�t+ 1tD(x; t)

1− A(x; t) .

Recall next that the product

q(x; t)= p′(x; t)p′′(x; t)is the generating function of the sequence of polynomials{qn(x)}n�0 defined by

qn(x)=n∑j=0

p′j (x)p

′′n−j (x), n= 0,1, . . . ,

i.e. the convolution of the sequences{p′n(x)}n�0 and{p′′

n(x)}n�0 which will be denotedby {(p′

n ∗ p′′n)(x)}n�0. Thus we have proved the next theorem.

Theorem 4.1. If the sequences of polynomials{p′n(x)}n�0 and{p′′

n(x)}n�0 belong to theclassesPk(a,b′) andPk(a,b′′) respectively, then the sequence{(p′

n ∗p′′n)(x)}n�0 belongs

to the classPk(a,d) where

di = b′i + b′′

i + iai, i = 1,2, . . . , k.

Of special interest is the case when the same sequence of polynomials{pn(x)}n�0 isconvoluted by itself many times thereof obtaining ther-fold convolution

p(r)n (x)= (pn ∗ pn ∗ · · · ∗ pn︸ ︷︷ ︸r times

)(x).

A recursive formula for definingp(r)n (x) is the following:

p(r)n (x)=pn(x), r = 1,n∑j=0

p(r−1)j (x)pn−j (x), r = 2,3, . . . . (4.1)

By repeated application of Theorem 4.1 we derive the following result.

Corollary 4.1. If the sequence of polynomials{pn(x)}n�0 belongs to the classPk(a,b),then the sequence of r-fold convolutions{p(r)n (x)}n�0 belongs to the classPk(a,d), with

di = (r − 1)iai + rbi, i = 1,2, . . . , k. (4.2)

26 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Corollary 4.1 can be used to establish an alternative recurrence relation for ther-fold con-volution polynomialsp(r)n (x), n=0,1, . . . .More specifically, we have the next interestingoutcome.

Corollary 4.2. If the sequence of polynomials{pn(x)}n�0 belongs to the classPk(a,b),then the r-fold convolution polynomialsp(r)n (x), n = 0,1, . . . satisfy the recurrence rela-tions

p(r)n (x)=1

n

k∑i=1([n+ i(r − 1)]ai + rbi)p(r)n−i (x), n�1. (4.3)

Proof. It suffices to observe that the sequence of polynomials{p(r)n (x)}n�0 belongs to theclassPk(a,d), with thedi , i = 1,2 . . . , k given by (4.2) and apply the recurrence scheme(2.1) withbi replaced bydi . �

It is perhaps unnecessary to point out that the computation of the sequence ofr-foldpolynomialsp(r)n (x), n = 0,1, . . . via (4.3) is much more tractable, as compared to therecurrence (4.1).For the benefit of the practical minded reader we use this occasion to illustrate the

connection between a few well-known classes of polynomials and, at the same time, re-establish efficient recursive relations for them the easy way. Let us first consider once againthe Legendre polynomialspn(x), n= 0,1, . . . defined in (2.2). Since{pn(x)}n�0 belongsto the classP2(a,b), with ai , bi given by (2.3) we can apply formula (3.4) or (3.5) to obtainthe corresponding generating functionp(x; t) as

p(x; t)= exp{∫ t

0

xs − s2s[1− (2xs − s2)] ds

}= 1√

1− 2xt + t2 .

Therefore, the 2r-fold convolution polynomials will have the following generating func-tion:

∞∑n=0

p(2r)n (x)tn = [p(x; t)]2r = 1

(1− 2xt + t2)r

which coincides with the generating function of the classical Gegenbauer polynomialsCrn(x), n= 0,1, . . . (see e.g.Chichara (1978)). Direct application of Corollary 4.2 revealsthatCrn(x) obey the well-known recursive relation

nCrn(x)= 2x(n+ r − 1)Crn−1(x)− (n+ 2r + 2)Crn−2(x), n�1.

It goeswithout saying that forr=1 the aforementioned generating function and recursivescheme reduce to the corresponding results for the Chebyshev polynomials of the secondkind.

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3927

As a second illustration, let us consider the shifted Fibonacci polynomials of orderk,pn(x), n= 0,1, . . . as defined by (2.5). Again by substitution of

A(x; t)=k∑i=1

aiti =

k∑i=1xti, B(x; t)=

k∑i=1

biti = 0

(see also (2.4)) in formula (3.5) we obtain

p(x; t)=(1− x

k∑i=1

t i

)−1

and applying Corollary 4.2 we may readily establish the next recurrence for ther-foldconvolution polynomialspn(x), n= 0,1, . . .:

np(r)n (x)=k∑i=1

x(n+ i(r − 1))p(r)n−i (x), n�1. (4.4)

It is worth mentioning that for the sequence of polynomials{F (k)n,r (x)}n�0 defined by

F (k)n,r (x)={0, n�r − 1,p(r)n−r (x), n�r

(see also (2.6)) the corresponding generating function takes on the form

∞∑n=0

F (k)n,r (x)tn = t r

(1− x

k∑i=1

t i

)−r

while recurrence (4.4) yields

F (k)n,r (x)=x

n− rk∑i=1

[n− r + i(r − 1)]F (k)n−i,r (x), n�r + 1.

This class of polynomials was first introduced byPhilippou et al. (1985), while the lastrecurrence relation was derived inPhilippou and Georghiou (1989)by a different method.We are now going to prove that the classPk is closed under the operation of partial

sum formulation. Before starting that, let us first establish a slightly more general result,described in the following theorem.

Theorem 4.2. Let{pn(x)}n�0beasequenceofpolynomialsbelonging to theclassPk(a,b),and{qn(x)}n�0 a new sequence of polynomials defined as

qn(x)=n∑j=0

(�+ j − 1

j

)gj (x)pn−j (x), n= 0,1, . . . ,

28 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

where g(x) is any polynomial in x and� is any positive integer. Then, the sequence{qn(x)}n�0 belongs to the classPk+1(a′,b′),wherea′

i=a′i (x),b

′i=b′

i (x), i=1,2, . . . , k+1are given by

a′i =

{a1+ g(x), i = 1,ai − g(x)ai−1, i = 2,3, . . . , k−akg(x), i = k + 1,

,

b′i =

{(�− 1)g(x)+ b1, i = 1,g(x)[(1− �)ai−1− bi−1] + bi, i = 2,3, . . . , k,g(x)[(1− �)ak − bk], i = k + 1.

Proof. It is straightforward that the generating functionq(x; t) of the sequence{qn(x)}n�0takes on the form

q(x; t)=(

1

1− tg(x))�p(x; t),

wherep(x; t) is the generating function of{pn(x)}n�0. Recalling formula (3.4) andmakinguse of the obvious expression(

1

1− tg(x))�

= exp[−� ln(1− tg(x))] = exp{∫ t

0

�g(x)

1− sg(x) ds}

we may write

q(x; t)= exp{∫ t

0

(�g(x)

1− sg(x) + C(x; s)s[1− A(x; s)]

)ds

},

where

C(x; s)= s �A(x; s)�s

+ B(x; s).

The term inside the integral can be expressed as

s�g(x)(1− A(x; s))+ (1− sg(x))C(x; s)s(1− [A(x; s)+ sg(x)(1− A(x; s))]) = C′(x; s)

s[1− A′(x; s)] ,

whereA′(x; s), C′(x; s) are polynomials insof degree at mostk + 1 (with at least one ofthembeing a polynomial insof degreek+1), i.e.q(x; t) is consistent with expression (3.4).This fact proves that{qn(x)}n�0 belongs to the classPk+1(a′,b′). The relation betweenai ,bi anda′

i , b′i can be easily established by considering the coefficients oft i in the identities

A′(x; t)=k+1∑i=1

a′i (x)t

i = tg(x)− tg(x)A(x; t)+ A(x; t)

= (a1+ g(x))t +k∑i=2(ai − g(x)ai−1)t i − akg(x)tk+1

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3929

and

B ′(x; t)=k+1∑i=1

b′i (x)t

i = C′(x; t)− t �A′(x; t)�t

= t�g(x)(1− A(x; t))+ (1− tg(x))C(x; t)− t �

�t[A(x; t)+ tg(x)(1− A(x; t))]

= t�g(x)(1− A(x; t))+ (1− tg(x))(t�A(x; t)

�t+ B(x; t)

)

− t((1− tg(x)) �A(x; t)

�t+ g(x)(1− A(x; t))

)= t�g(x)(1− A(x; t))+ (1− tg(x))B(x; t)− tg(x)(1− A(x; t))= t (�− 1)g(x)(1− A(x; t))+ (1− tg(x))B(x; t). �

Applying Theorem 4.2 in the special case� = 1, g(x) = 1 we immediately deduce thenext corollary pertaining to partial sums.

Corollary 4.3. Let {pn(x)}n�0 be a sequence of polynomials belonging to the classPk(a,b), and{qn(x)}n�0 a new sequence of polynomials defined as

qn(x)=n∑i=0

pn−i (x), n= 0,1, . . . .

Then, the sequence{qn(x)}n�0 belongs to the classPk(a′,b′),wherea′

i=a′i (x), b

′i=b′

i (x),i = 1,2, . . . , k + 1 are given by

a′i =

{a1+ 1, i = 1,ai − ai−1, i = 2,3, . . . , k,−ak, i = k + 1,

b′i =

{b1, i = 1,bi − bi−1, i = 2,3, . . . , k,−bk, i = k + 1.

As an illustration let us consider the sequence of polynomials{qn(x)}n�0 defined by

qn(x)=n∑i=0

pn−i (x), n= 0,1, . . . ,

wherepn(x), n=0,1, . . . are the shifted Fibonacci polynomials introduced by (2.4), (2.5).By virtue of Corollary 4.3 we may state that the sequence of partial sums{qn(x)}n�0belongs to the familyPk+1(a′,b′), whereb′

i = b′i (x)= 0, i = 1,2, . . . , k + 1, and

a′i = a′

i (x)={x + 1, i = 1,0, i = 2,3, . . . , k,−x, i = k + 1.

30 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Consequently, the following recurrence relation holds true:

qn(x)=k+1∑i=1

(a′i +

b′i

n

)qn−i (x)= (x + 1)qn−1(x)− xqn−k−1(x), n�1.

Closing this section, we establish the following interesting result, which will be provedquite useful in the next section.

Theorem 4.3. Every finite sequence of polynomials{pn(x)}n=0,1,...,k belongs to the classPk(a,b), whereai = ai(x), bi = bi(x) are given by

ai = −pi(x), bi = 2ipi(x), 1� i�k. (4.5)

Proof. In this special case we have

p(x; t)=k∑i=0

pi(x)ti = 1+

k∑i=1

pi(x)ti

and hence

1

p(x; t)�p(x; t)

�t=∑ki=1 ipi(x)t i−1

1+∑ki=1pi(x)t i

=∑ki=1[i(−pi(x))+ 2ipi(x)]t i−11−∑k

i=1(−pi(x))t i.

It is now clear that

�p(x; t)�t

= p(x; t)∑ki=1(iai + bi)t i−11−∑k

i=1 ait i

with ai andbi defined as in (4.5) and the required result is easily established by Theorem3.1 (see also the discussion following the proof of this theorem).�

It is noteworthy that in Theorem 4.3 the classPk(a,b), does not satisfy the condition thatai = ai(x), bi = bi(x), i = 1,2, . . . , k, are polynomials inx of degree at most 1; however,as already stated earlier, violation of this assumption does not affect the results proved inthe previous section.

5. Probability applications

Let us consider an infinite sequence of iid binary outcomesX1, X2, . . . (Bernoulli trials)with success probabilitiesp = P(Xi = 1) and failure probabilitiesq = P(Xi = 0) =1 − p, i = 1,2, . . . . Let alsoTr denote the waiting time for therth occurrence of aneventE (simple or composite) in the sequenceX1, X2, . . . , that is the number of trialsup to and including therth appearance ofE. It is assumed that, with probability 1,Eoccurs at least once in an infinitely prolonged sequence of trials. If we denote bySn the

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3931

number of successes in the firstn trials, the probabilitymass function ofTr maybeexpressedas

f (n)= P(Tr = n)=n∑i=0

P(Tr = n|Sn = n− i)P (Sn = n− i)

=n∑i=0

P(Tr = n|Sn = n− i)(ni

)pn−iqi .

SinceSn is a sufficient statistic forp, it is clear that

�i =(ni

)P(Tr = n|Sn = n− i), i = 0,1, . . . , n

does not depend onp. As a consequence,f (n) takes on the form

f (n)= P(Tr = n)= pnpn(q/p), (5.1)

where

pn(x)=n∑i=0

�ixi , n= 0,1, . . .

is a polynomial inx of degree at mostn. Let us also denote by

GTr (p; z)= E(zTr )the probability generating function ofTr .If the support of the random variableTr is {m,m+1, . . .} (m is a given positive integer),

then the shifted random variableX=Tr −m has support{0,1, . . .} and its probability massfunction will be given by

P(X = n)= P(Tr =m+ n)= f (m+ n)= pm+npm+n(q/p)= pm+nhn(q/p), n= 0,1, . . . , (5.2)

where

hn(x)= pn+m(x), n= 0,1, . . .is a “shifted” sequence of polynomials. It is of interest to note that the probability generatingfunction of the random variableXmay be written as

z−mE(zTr )= E(zX)=∞∑n=0

P(X = n)zn

and replacingP(X = n) by virtue of (5.2) we deduce

z−mGTr (p; z)=∞∑n=0

pm+nhn(q/p)zn.

32 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Hence

GTr (p; z)= (pz)m∞∑n=0

hn(q/p)(pz)n = (pz)mH(q/p;pz),

where

H(x; t)=∞∑n=0

hn(x)tn

is the generating function of the sequence of polynomials{hn(x)}n�0. On using the nota-tionsx = q/p, t = pz we can readily establish the next useful expression

H(x; t)= t−mGTr(

1

1+ x ; t (1+ x)). (5.3)

This formula offers a very efficient tool for calculating the generating function of thesequenceof shiftedpolynomialshn(x),n=0,1, . . .when theprobability generating functionof Tr is available.Furthermore, let us denote byFTr (p; z) the generating function of the cumulative distri-

bution functionF(n)= P(Tr�n) of Tr . SinceFTr (p; z)= 1

1− z GTr (p; z)we may write, by virtue of (5.3),

1

1− t (x + 1) H(x; t)= t−mFTr

(1

1+ x ; t (1+ x)).

We are now in possession of all necessary ingredients to prove the following interestingresults.

Theorem 5.1. If the sequence of polynomials{hn(x)}n�0 belongs to the classPk(a,b)then the probability mass functionf (n) of the waiting time random variableTr obeys therecurrence

f (n)=k∑i=1

(ai(q/p)+ bi(q/p)

n−m)pif (n− i).

Proof. Let us first write down the recursive relation

hn(x)=k∑i=1

(ai(x)+ bi(x)

n

)hn−i (x)

which characterizes the classPk(a,b), and replacehn(x),hn−i (x) bypm+n(x),pm+n−i (x)respectively. The resulting relation, on settingx = q/p, and changingn to n−m yields

pn(q/p)=k∑i=1

(ai(q/p)+ bi(q/p)

n−m)pn−i (q/p)

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3933

and the result we are looking at is easily deduced if wemultiply both sides of the last identityby pn and make use of (5.1).�

Theorem 5.2. If the sequence of polynomials{hn(x)}n�0 belongs to the classPk(a,b),the cumulative distribution functionF(n) of the waiting time random variableTr obeys therecurrence

F(n)= P(Tr�n)=k+1∑i=1

(a′i (q/p)+

b′i (q/p)

n−m)piF (n− i),

wherea′i = a′

i (x), b′i = b′

i (x), i = 1,2, . . . , k + 1 are given by

a′i =

{a1+ x + 1, i = 1,ai − (x + 1)ai−1, i = 2,3, . . . , k,−(x + 1)ak, i = k + 1,

b′i =

{b1, i = 1,bi − (x + 1)bi−1, i = 2,3, . . . , k,−(x + 1)bk, i = k + 1.

Proof. Let {qn(x)}n�0 be a sequence of polynomials with generating function

Q(x; t)= 1

1− t (x + 1) H(x; t).

Then

qn(x)=n∑j=0

(x + 1)jhn−j (x), n= 0,1, . . . (5.4)

and for�= 1 andg(x)= x + 1, Theorem 4.2 gives

qn(x)=k+1∑i=1

(a′i (x)+

b′i (x)

n

)qn−i (x), n= 0,1, . . . , (5.5)

where

a′1= a1+ (x + 1), a′

k+1= −(x + 1)ak, b′1= b1, b′

k+1= −(x + 1)bk,a′i = ai − (x + 1)ai−1, b′

i = bi − (x + 1)bi−1 for i = 2,3, . . . , k.

Insertion of (5.4) into (5.5) yields

n∑j=0

(x + 1)jhn−j (x)=k+1∑i=1

(a′i (x)+

b′i (x)

n

) n−i∑j=0

(x + 1)j hn−i−j (x)

34 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

and replacinghn−j (x), hn−i−j (x) by pn+m−j (x), pn+m−i−j (x) respectively, we get, onsettingx = q/p,

n∑j=0

p−jpn+m−j (q/p)=k+1∑i=1

(a′i (q/p)+

b′i (q/p)

n

) n−i∑j=0

p−jpn+m−i−j (q/p).

Finally, if we multiply both sides of the last identity bypn+m, make use of (5.1) and useformulaF(n) = ∑n

j=m f (j), we get

F(n+m)=k+1∑i=1

(a′i (q/p)+

b′i (q/p)

n

)piF (n+m− i).

The proof of the theorem follows easily by replacingn by n−m. �

Wenowproceed to review a few typical exampleswhereTheorems 5.1, 5.2 and the theorypresented in the previous sections can be beneficially used.

Example 1. Let us first consider the casewhere the interest focuses on the non-overlappingoccurrences of success runs of fixed lengthk�2. For a detailed study of success runsproblems and their applications to several research areas the reader is referred to the recentmonograph byBalakrishnan and Koutras (2002). If Tr denotes the waiting time for therthoccurrence of a (non-overlapping) success run of lengthk in the sequenceX1, X2, . . . theprobability generating function ofTr will be given by

GTr (p; z)= E(zTr )=(

(pz)k

1− qz∑k−1i=0 (pz)i

)r. (5.6)

Since the support of the random variableTr is {rk, rk + 1, . . .}, a direct application of(5.3) results in the following expression for the generating functionH(t; x) of the sequenceof polynomials{hn(x)}n�0 associated with the shifted random variableX = Tr − rk (seealso (5.2)):

H(x; t)= t−rkGTr(

1

1+ x ; t (1+ x))

=(1− x

k∑i=1

t i

)−r.

As already indicated in the secondexample followingCorollary 4.2, theRHScorrespondstoPk(a,b), with ai(x)= x, bi(x)= xi(r−1), i=1,2, . . . , k and Theorem 5.1 reveals thatf (n)= P(Tr = n) satisfy the recurrence relation

f (n)= 1

n− rkk∑i=1

[n− rk + i(r − 1)]qpi−1f (n− i), n�rk + 1.

This formula was first given byPhilippou and Georghiou (1989), while Charalambides(1986)established a similar recurrence making use of truncated Bell polynomials.

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3935

Moreover, by exploiting Theorem 5.2 we may arrive at the following recursive schemefor the cumulative distribution functionF(n)= P(Tr�n) of Tr :

F(n)=(1+ q + q(r − 1)

n− rk)F(n− 1)− qpk−1

(1+ k(r − 1)

n− rk)F(n− k − 1)

− qk∑i=2

(q − (r − 1)(1− iq)

n− rk)pi−2F(n− i).

An alternative expression forf (n) can be derived by writingGTr (p; z) in the form

GTr (p; z)=(

(pz)k(1− pz)1− z+ (qz)(pz)k

)r= (pz)rk(1− pz)r

(1

1− z+ qpkzk+1)r(5.7)

and making use of the sequence of polynomials{gn(x)}n�0 with generating function givenby

G(x; t)=∞∑n=0

gn(x)tn =

(1

1− (1+ x)t + xtk+1)r.

Manifestly,gn(x), n = 0,1, . . . can be viewed asr-fold convolutions of the polynomialscorresponding to the classPk+1(a,b), with

a1= x + 1, ak+1= −x, a2= · · · = ak = 0, b1= b2= · · · = bk+1= 0.By Corollary 4.2gn(x), n= 0,1, . . . satisfy the recurrence relations

ngn(x)= (1+ x)(n+ r − 1)gn−1(x)− x[n+ (k + 1)(r − 1)]gn−k−1(x), n�1 (5.8)

and therefore the evaluation off (n)=P(Tr = n) can be easily carried out by the aid of theexpression

f (n)= pnr∑i=0(−1)i

( ri

)gn−rk−i (q/p), n�rk

(f (n)= 0 for n< rk ) which results immediately form the identity∞∑n=0

f (n)zn =GTr (p; z)= (pz)rk(1− pz)rG(q/p;pz)

= (pz)rk(

r∑i=0(−1)i

( ri

)(pz)i

) ∞∑n=0

gn(q/p)(pz)n.

Example 2. As a second example let us consider the waiting timeT ′r for therth occurrence

of a success run of length at leastk. The probability generating function ofT ′r is given by

(see e.g.Balakrishnan and Koutras, 2002)

GT ′r(p; z)= E(zT ′

r )=GTr (p; z)(

qz

1− pz)r−1

, (5.9)

36 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

whereGTr (p; z) is as in (5.7). Expressing the RHS in the form

GT ′r(p; z)= (pz)rk(qz)r−1(1− pz)

(1

1− z+ qpkzk+1)r

and taking into account the aforementioned analysis forTr , we may write

∞∑n=0

P(T ′r = n)zn = (pz)rk(qz)r−1(1− pz)

∞∑n=0

gn(q/p)(pz)n

and the following expression follows immediately (by picking out the coefficients ofzn inboth sides)

P(T ′r = n)=

{0, n< rk + r − 1,qr−1pn−r+1[gn−rk−r+1(q/p)− gn−rk−r (q/p)], n�rk + r − 1.

This formula, in conjunction with the recurrence (5.8), offers an efficient computationalscheme for the evaluation of the probability mass function ofT ′

r .It is worth mentioning that the generating function ofT ′

r can be expressed in the form(see (5.6) and (5.9))

GT ′r(p; z)= (pz)rk(qz)r−1

(1

1− pz)r−1( 1

1− qz∑k−1i=0 (pz)i

)r

and making use of the sequence of polynomials{qn(x)}n�0 with generating function

Q(x; t)=∞∑n=0

qn(x)tn = (1− t)1−r

(1− x

k∑i=1

t i

)−r= (1− t)1−rH(x; t)

we may write

GT ′r(p; z)=

∞∑n=0

P(T ′r = n)zn = (pz)rk(qz)r−1Q(q/p;pz).

Therefore, the evaluation ofP(T ′r = n) can also carried out by the aid of the expression

P(T ′r = n)=

{0, n< rk + r − 1,qr−1pn−r+1qn−rk−r+1(q/p), n�rk + r − 1

with the polynomials{qn(x)}n�0 being themembers of the classPk+1(a′,b′) (cf. Theorem4.2 for�= r − 1 andg(x)= 1) with

a′1= x + 1, a′

k+1= −x, a′2= · · · = a′

k = 0,b′1= (x + 1)(r − 1)− 1, b′

k+1= −x(r − kr − k),b′i = −x for i = 2,3, . . . , k.

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3937

Example 3. Asafinal example, let usconsider thewaiting timeT (2,m)r for therthoccurrenceof a generalized run of type 2/m. The probability generating function ofT (2,m)r is given by(see e.g.Balakrishnan and Koutras, 2002)

GT(2,m)r

(p; z)= (pz)2r (1− (qz)m−1)r(

1

1− qz− (pz)(qz)m−11

1− qz)r.

In order to proceed to the investigation of the distribution ofT (2,m)r we introduce thesequence of polynomials{pn(x)}n�0 and{qn(x)}n�0 with generating functions given by

H(x; t)= 1

1− xt − xm−1tm, Q(x; t)= 1

1− xt H(x; t)

respectively. Since{pn(x)}n�0 belongs to the classPm(a,b), with

a1= x, am = xm−1, a2= · · · = am−1= 0, b1= b2= · · · = bm = 0it follows from Theorem 4.2 that{qn(x)}n�0 belongs to the classPm+1(a′,b′), with

a′1= 2x, a′

2= x2, a′m = xm−1, a′

m+1= −xm, a′2= · · · = a′

m−1= 0,b′1= b′

2= · · · = b′m+1= 0.

A suitable application of Corollary 4.2 reveals that ther-fold convolutionsq(r)n (x) ofqn(x) satisfy the recurrence relation

nq(r)n (x)= 2x(n+ r − 1)q(r)n−1(x)− x2(n+ 2(r − 1))q(r)n−2(x)+ xm−1(n+m(r − 1))q(r)n−m(x)− xm(n+ (m+ 1)(r − 1))q(r)n−m−1(x).

Theevaluation off (n)=P(T (2,m)r =n) can thenbe carried out by the aid of the expression

f (n)= pnr∑i=0(−1)i

( ri

)(q/p)(m−1)iq(r)n−2r−(m−1)i (q/p), n�2r

(f (n)= 0 for n<2r) which results immediately form the identity∞∑n=0

f (n)zn =GT(2,m)r

(p; z)= (pz)2r (1− (qz)m−1)rQr(q/p;pz)

= (pz)2r(

r∑i=0(−1)i

( ri

)(qz)(m−1)i

) ∞∑n=0

q(r)n (q/p)(pz)n.

In closing this section we mention that the theory presented in the previous sectionsremains valid if the polynomialspn(x),n=0,1, . . . are of degree 0. In this case the sequence{qn(x)}n�0 reduces to a (recursively defined) sequence of numbers and the investigationof the associated distribution becomes even simpler.We now proceed to review two examples where the above remark applies. Note that,

sincepn(x), n= 0,1, . . . are now pure numbers, the variablex has been dropped from thenotation used hereafter.

38 D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–39

Consider first the case of a random variableX following a discrete distribution with finitesupport{0,1, . . . , k}, i.e.

P(X = n)= �n, n= 0,1, . . . , k,and probability generating function

GX(t)=k∑n=0

P(X = n)tn =k∑n=0

�ntn.

Denote byXr ther-fold convolution ofX and letGXr (t) be its generating function, i.e.

GXr (t)=rk∑n=0

P(Xr = n)tn =(

k∑n=0

�ntn)r

= �r0

(k∑n=0

�n�0tn

)r= �r0

(k∑n=0

pntn

)r

with pn= �n/�0, n=0,1, . . . , k. According to Theorem 4.3 the finite sequence of numbers{pn}n=0,1,...,k belongs to the classPk(a,b), with

ai = − �i�0, bi = 2i �i

�0, i = 1,2, . . . , k

and therefore the sequence of numbersp(r)n (the r-fold convolution ofpn) will satisfy therecurrence relation (see Corollary 4.2)

p(r)n = 1

n�0

k∑i=1(i(r + 1)− n)�ip(r)n−i .

Hence, the evaluation ofP(Xr = n) can be carried out by the aid of the expressionP(Xr = n)= �r0p

(r)n , 0�n�rk.

Let us next consider the case of a random variableX following the extended (or multipa-rameter) Poisson distribution of orderk (seeAki, 1985; Philippou, 1988) with probabilitygenerating function

GX(t)=∞∑n=0

P(X = n)tn = exp{

−k∑i=1

�i

}exp

{k∑i=1

�i t i}.

It can be shown that the sequence of numbers{pn}n�0 with generating function

p(t)=∞∑n=0

pntn = exp

{k∑i=1

�i t i}

belongs to the classPk(a,b), with

a1= a2= · · · = ak = 0, bi = i�i , i = 1,2, . . . , k.

D.L. Antzoulakos, M.V. Koutras / Journal of Statistical Planning and Inference 135 (2005) 18–3939

Therefore, the evaluation ofP(X = n) can be carried out by the aid of the expression

P(X = n)= exp{

−k∑i=1

�i

}pn, n�0

or equivalently, by the recursive scheme

P(X = n)= 1

n

k∑i=1

i�iP (X = n− i), n�1.

Furthermore, manipulating over Theorem’s 4.2 outcome, we may verify that the cumu-lative distribution functionF(n) of X satisfies the recursive scheme

nF(n)= (n+ �1)F (n− 1)+k∑i=2(i�i − (i − 1)�i−1)F (n− i)

− k�kF (n− k − 1), n�1.

Note that for�1 = a�, �2 = a2/2, �3 = �4 = · · · = �k = 0 the distribution ofX reducesto the well-known Hermite distribution (see e.g.Johnson et al., 1992).

References

Aki, S., 1985. Discrete distributions of orderk on a binary sequence. Ann. Inst. Statist. Math. 37, 205–224.Balakrishnan, N., Koutras, M.V., 2002. Runs and Scans with Applications. Wiley, NewYork.Charalambides, Ch.A., 1986. On discrete distributions of orderk. Ann. Inst. Statist. Math. 38, 557–568.Chichara, T.S., 1978. An Introduction to Orthogonal Polynomials. Gordon and Breach, NewYork.Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Discrete Univariate Distributions. second ed. Wiley, NewYork.Lebedev, N.N., 1965. Special Functions and Their Applications. Prentice-Hall, Englewood Cliffs, NJ.Philippou, A.N., 1988. On multiparameter distributions of orderk. Ann. Inst. Statist. Math. 40, 467–475.Philippou, A.N., Antzoulakos, D.L., 1990. Multivariate Fibonacci polynomials of orderk and the multiparameternegative binomial distribution of the same order. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (Eds.),Applications of Fibonacci Numbers, vol. 3. Kluwer Academic Publishers, Dordrecht, pp. 273–279.

Philippou, A.N., Georghiou, C., 1989. Convolutions of Fibonacci-type polynomials of orderk and the negativebinomial distribution of the same order. Fibonacci Quart. 27, 209–216.

Philippou, A.N., Georghiou, C., Philippou, G.N., 1985. Fibonacci-type polynomials of orderk with probabilityapplications. Fibonacci Quart. 23, 100–105.

Sansone, G., 1959. Orthogonal Functions. Interscience Publ., NewYork.Szegö, G., 1975. Orthogonal Polynomials. fourth ed. American Mathematical Society, Providence, RI.