# Certain Class of Generating Functions for the Incomplete Hypergeometric Functions

Authors: Junesang Choi; Praveen Agarwal
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. In this paper, we aim to establish certain generating functions for the incomplete hypergeometric functions introduced by Srivastava et al. (2012). All the derived results in this paper are general and can yield a number of (known and new) results in the theory of generating functions.


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## Body

## 1. Introduction and Definitions

A lot of research work has recently come up on the study and development of the familiar incomplete Gamma type functions likeγ(s,x) and Γ(s,x) given in (1) and (2), respectively. The study of incomplete Gamma functions has a very long history (see, e.g., [1]) and now stands on fairly firm footing through the research contributions of various authors (see, e.g., [2–17]). Incomplete Gamma functions are important special functions and their closely related ones are widely used in physics and engineering; therefore, they are of interest to physicists, engineers, statisticians, and mathematicians.The theory of the incomplete Gamma functions, as a part of the theory of confluent hypergeometric functions, has received its first systematic exposition by Tricomi [18] in the early 1950s. The familiar incomplete Gamma functions γ(s,x) and Γ(s,x) are defined, respectively, by
(1)γs,x≔∫0xts-1e-tdt,Rs>0;x≥0,(2)Γs,x≔∫x∞ts-1e-tdt,x≥0;Rs>0whenx=0.The following decomposition formula holds:(3)γs,x+Γs,x=Γs,Rs>0,
where Γ(s) is the familiar Gamma function defined by
(4)Γs≔∫0∞ts-1e-tdt,Rs>0.Historically, (1) and (2) were first studied in 1877 for x=1 by Prym [1]. The functions (1) and (2) are also referred to as Prym’s functions. For general x>0 (even for x<0), the function (2) appears in Exercises de Calcul Integral by Legendre [19] and in some of his later works.The functionΓ(s,x) can be expressed in terms of Tricomi’s confluent hypergeometric function Ψ(a,c;x) as follows (see [6, page 266, Equation 6.9.2(21)]):
(5)Γs,x≔e-xΨ1-s,1-s;x.In terms of the Gamma functionΓ(z), Pochhammer symbol (λ)n is defined (for λ∈C) by (see, e.g., [10, page 2 and pages 4–6])
(6)λn≔1,n=0λλ+1⋯λ+n-1,n∈N≔1,2,3,…:=Γλ+nΓλ,λ∈C∖Z0-,
where C and Z0- denote the sets of complex numbers and nonpositive integers, respectively.Recently, Srivastava et al. [16] introduced and studied some fundamental properties and characteristics of a family of the following two potentially useful generalized incomplete hypergeometric functions defined as follows:
(7)pγqa1,x,a2,…,ap;xxxxxxb1,…,bq;z≔∑n=0∞a1;xna2n⋯apnb1n⋯bqn·znn!,(8)pΓqa1,x,a2,…,ap;xxxxxxb1,…,bq;z≔∑n=0∞a1;xna2n⋯apnb1n⋯bqn·znn!,
where (a1;x)n and [a1;x]n are certain interesting generalizations of the Pochhammer symbol (λ)n which are defined, in terms of the incomplete Gamma type functions γ(s,x) and Γ(s,x) given in (1) and (2), by
(9)λ;xν≔γλ+ν,xΓλ,λ,ν∈C;x≥0,λ;xν≔Γλ+ν,xΓλ,λ,ν∈C;x≥0.These incomplete Pochhammer symbols(λ;x)ν and [λ;x]ν, which were defined by Srivastava et al. [16], like (3), also satisfy the following decomposition relation:
(10)λ;xν+λ;xν=λν,λ,ν∈C;x≥0.Remark 1.
As already mentioned by Srivastava et al. [16, Remark 7] (see also [17, page 3220, Remark]), since
(11)λ;xn≤λn,λ;xn≤λn,n∈N;λ∈C;x≥0,
the precise (sufficient) conditions under which the infinite series in the definitions (7) and (8) would converge absolutely can be derived from those that are well-documented in the case of the generalized hypergeometric function pFq (p,q∈N) (see, for details, [20, pages 72-73] and [11, page 20]; see also [21–23]). Indeed, in their special case when x=0, both pγq (p,q∈N) and pΓq (p,q∈N) would reduce immediately to the extensively investigated generalized hypergeometric function pFq (p,q∈N) (see, e.g., [20, Chapter 5]; see also [10, Section 1.5]). Furthermore, as an immediate consequence of the definitions (7) and (8), we have the following decomposition formula:
(12)pγqa1,xa2,…,ap;b1,…,bq;z+pΓqa1,x,a2,…,ap;b1,…,bq;z=pFqa1,…,ap;b1,…,bq;z,
in terms of the familiar generalized hypergeometric function pFq (p,q∈N).Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. They are used in finding certain properties and formulas for numbers and polynomials in a wide variety of research subjects, indeed, in modern combinatorics. For a systematic introduction to, and several interesting (and useful) applications of, the various methods of obtaining linear, bilinear, bilateral, or mixed multilateral generating functions for a fairly wide variety of sequences of special functions (and polynomials) in one, two, and more variables, among much abundant literature, we refer to the extensive works by Srivastava and Manocha [24] and Agarwal and Koul [25]. In this regard, in fact, a remarkably large number of generating functions involving a variety of special functions have been developed by many authors (see, e.g., [24, 26]; see also [27]). Also many generating functions containing the incomplete hypergeometric functions (7) and (8) have been presented (see, e.g., [17, Corollary 3]). Here, motivated mainly by the works of both Chen and Srivastava [28] and Srivastava and Cho [17], we present certain generating functions involving the incomplete hypergeometric functions (7) and (8). Furthermore, it should be mentioned in passing that our results in the present paper are established by using a different method employed by [17].


## 2. Generating Functions for the Incomplete Hypergeometric Functions

In this section, we establish certain generating functions for the incomplete hypergeometric functions (7) and (8) asserted by Theorem 2.Theorem 2.
The following generating functions hold true:(13)∑k=0∞α+k-1kpγq+1a1,x,a2,…,ap;1-α-k,b1,…,bq;ztk=1-t-αpγq+1a1,x,a2,…,ap;xx1-α,b1,…,bq;z1-t,xxxxxxxxxxxxxxxxxixxx≥0;α∈C;t<1,(14)∑k=0∞α+k-1kpΓq+1xa1,x,a2,…,ap;1-α-k,b1,…,bq;ztk=1-t-αpΓq+1a1,x,a2,…,ap;xx1-α,b1,…,bq;z1-t,xxxxxxxxxxxxxxxixxxxxxx≥0;α∈C;t<1.Proof.
For convenience, let the left-hand side of (13) be denoted by S. Applying the series expression of (7) to S, we get
(15)S=∑k=0∞α+k-1k·∑n=0∞a1;xna2n⋯apn1-α-knb1n⋯bqn·znn!tk.
Using the following known identities (see, e.g., [10, page 5]):
(16)αn=Γα+1n!Γα-n+1,n∈N0;α∈C,Γα-nΓα=-1n1-αn,n∈N0;α∈C∖Z,Z being the set of integers and N0≔N∪{0}, we can prove the following identity (see [28, page 169]):
(17)1-α-kn=1-αnα+k-1k·α-n+k-1k-1,k,n∈N0.
By changing the order of summations in (15) and using the identity (17), after little simplification, we have
(18)S=∑n=0∞a1;xna2n⋯apn1-αn,b1n⋯bqn·znn!∑k=0∞α-n+k-1ktk.
We find that the inner sum in (18) is the generalized binomial expansion
(19)∑k=0∞α-n+k-1ktk=1-t-α+n,t<1.
Finally, replacing the inner sum of (18) by the identity (19) yields our desired result (13).
It is easy to see that a similar argument as in the proof of (13) will establish the result (14). This completes the proof of Theorem 2.Remark 3.
Recently, Srivastava and Cho [17] presented a very general class of certain interesting generating functions involving the incomplete hypergeometric functions (7) and (8) by essentially using the following interesting and useful unified expansion formula given by Gould (see [29, page 196, Equation (9)]; see also [17, page 3221]):
(20)∑k=0∞κκ+β+1nα+β+1nntn=1+ξα∑n=0∞-1nα-κnn+κβ+1n-1ξ1+ξn,
where α, β, and κ are complex numbers independent of n and ξ is a function of t defined implicitly by
(21)ξ=t1+ξβ+1,ζ0=0.The results [17, Corollary 3] look very similar to those given in Theorem 2. Yet, it is easy to see that they cannot be special or general cases of the other one’s results.


## 3. Further Generalization of the Generating Functions for the Incomplete Hypergeometric Functions

A further generalization of the incomplete hypergeometric functions (7) and (8) is given in the following definition.Definition 4.
Let us introduce two sequences{Λk(α,β)(z)}k=0∞ and {Ωk(α,β)(z)}k=0∞ defined by
(22)Λkα,βz=Λkα,βa1,x,a2,…,ap;xxxxxxb1,…,bq;z=pγq+βxxxxxxxa1,x,a2,…,ap;Δβ;1-α-k,b1,…,bq;z,xxxxxxxxxxxxxxx≥0;β∈N;α∈C,Ωkα,βz=Ωkα,βa1,x,a2,…,ap;xxxxxxb1,…,bq;z=pΓq+βxxxxxxa1,x,a2,…,ap;Δβ;1-α-k,b1,…,bq;z,xxxxxxxxxxxxxx≥0;β∈N;α∈C,
where, for convenience, Δ(β;α) abbreviates the array of N parameters as follows:
(23)αβ,α+1β,…,α+β-1β,α∈C;β∈N.
Then, as in Theorem2, we can give the following generating functions for the generalized incomplete hypergeometric functions (22) asserted by Theorem 5.Theorem 5.
Each of the following identities holds true:(24)∑k=0∞α+m+k-1kΛm+kα,βztk=1-t-α-mΛmα,βz1-tβ,x≥0;m∈N0;α∈C;β∈N;t<1,(25)∑k=0∞α+m+k-1kΩm+kα,βztk=1-t-α-mΩmα,βz1-tβ,(26)x≥0;m∈N0;α∈C;β∈N;t<1.Proof.
Similarly as in Theorem2, we can prove the results in Theorem 5. So their details are left to the interested reader by, instead of the essential identity (17), presenting the following identity:
(27)1-α-m-kβn=1-α-mβnα+m+k-1k·α+m-βn+k-1k-1,xxxxk,n∈N0;β∈N.It should be noted that, if we setβ=1 and replace α by α-m in (24), we are easily led to the result (13).Concluding Remarks. If we add the two generating functions (13) and (14) and use the decomposition formula (11), we have an interesting result expressed in terms of generalized hypergeometric functions pFq:
(28)∑k=0∞α+k-1kpFq+1xxxxxa1,a2,…,ap;1-α-k,b1,…,bq;ztk=1-t-αpFq+1xxa1,a2,…,ap;1-α,b1,…,bq;z1-t,xxxxxxxxxxxxxxxxxxxxxxxα∈C;t<1.We also observe that the result (28) corresponds to that given in [28, page 170, Equation (5.12)].The generalized incomplete hypergeometric functions given in (7) and (8) reduce, when x=0, to the generalized hypergeometric function pFq (p,q∈N0) whose particular cases are known to express most of the special functions occurring in the mathematical, physical, and engineering sciences. Therefore most of the known and widely investigated special functions are expressible also in terms of the generalized incomplete hypergeometric functions pγq (p,q∈N0) and pΓq (p,q∈N0) (for some interesting examples and applications, see [16, Sections 5 and 6]). In view of this observation, the results presented here, being of general character, can yield numerous generating functions for a certain class of incomplete hypergeometric polynomials (see [17]) and other special functions which are expressible in terms of hypergeometric functions. Finally, we conclude our present investigation by remarking that our results presented here are also believed to give some contribution to the communication theory, probability theory, and groundwater pumping modeling.



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