KYUNGPOOK Math. J. 2019; 59(2): 293-299  
An Identity Involving Product of Generalized Hypergeometric Series 2F2
Yong Sup Kim, Junesang Choi, Arjun Kumar Rathie
Department of Mathematics Education, Wonkwang University, Iksan 54538, Republic of Korea
e-mail : yspkim@wonkwang.ac.kr

Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
e-mail : junesang@mail.dongguk.ac.kr

Department of Mathematics, Vedant College of Engineering & Technology (Rajasthan Technical University), Bundi, Rajasthan State, India
e-mail : arjunkumarrathie@gmail.com
* Corresponding Author.
Received: August 27, 2018; Revised: October 16, 2019; Accepted: October 16, 2018; Published online: June 23, 2019.
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Abstract

A number of identities associated with the product of generalized hypergeometric series have been investigated. In this paper, we aim to establish an identity involving the product of the generalized hypergeometric series 2F2. We do this using the generalized Kummer-type II transformation due to Rathie and Pogany and another identity due to Bailey. The result presented here, being general, can be reduced to a number of relatively simple identities involving the product of generalized hypergeometric series, some of which are observed to correspond to known ones.

Keywords: generalized hypergeometric series, Kummer-type I transformation, Kummer-type II transformation, Gauss’s summation theorem for 2F1(1), Watson summation theorem for 3F2
1. Introduction and Preliminaries

The generalized hypergeometric function pFq with p numerator and q denominator parameters is defined by (see, e.g., [2, 5, 12]; see also [16, Section 1.5])

Fpq[α1,,αp;β1,,βq;z]=Fpq[α1,,αp;β1,,βq;z]=n=0(α1)n(αp)n(β1)n(βq)nznn!,

where (λ)ν denotes the Pochhammer symbol defined (for λ, ν ∈ ℂ), in terms of the Gamma function Γ, by

(λ)ν:=Γ(λ+ν)Γ(λ)={1(ν=0;λ/{0})λ(λ+1)λ(λ+n-1)(ν=n;λ).

The well-known Kummer-type I transformation for the series 1F1 is (see, e.g., [12])

e-xF11[a;b;x]=F11[b-a;b;-x].

This result (1.3) was obtained by Kummer [7] and [8] who used the theory of differential equations. Bailey [1] derived this result by employing classical Gauss’s summation theorem (see, e.g., [12, p. 48]).

Paris [9] generalized (1.3) in the form

e-xF22[a,1+d;b,d;   x]=F22[b-a-1,f+1;b,f;   -x],

where

f:=d(a-b+1)a-d.

The particular case d=12 of (1.4) reduces to Exton’s result [4].

Kummer [7, 8] also used the theory of differential equations to obtain the following result

e-x2F11[a;2a;   x]=F01[-;a+12;   x216],

which is often referred to as Kummer-type II transformation.

Using Gauss’s summation theorem, the result (1.6) has been re-derived by Bailey [1] and Rathie and Choi [14] (see also [3]).

Motivated by the Kummer-type I transformation (1.4), Rathie and Pogany [15] also generalized the Kummer-type II transformation (1.6) in the form

e-x2F22[a,1+d;2a+1,d;   x]=F01[-;a+12;   x216]+x(2a-d)2d(2a+1)F01[-;a+32;   x216].

It is obvious that the case d = 2a of (1.7) reduces to (1.6).

Using the theory of differential equations, Preece [10] established the following interesting identity involving product of generalized hypergeometric series

F11[a;2a;   x]F11[a;2a;   -x]=F12[a;a+12,2a;   x24].

Equation (1.8) is a special case of [6, Exercise 7.24(b)]. In fact, Koepf [6, Example 7.3] showed how such identities can be derived by using the Zeilberger algorithm.

Using (1.3), we can express (1.8) in the form

{F11[a;2a;   x]}2=ex   F12[a;a+12,2a;   x24].

Rathie [13] proved Preece’s identity (1.9) by a very short method and obtained two results closely related to it.

By employing the classical Watson’s theorem on the sum of a 3F2 with unit argument (see, e.g., [2, p. 16]), Bailey [1] generalized Preece’s identity (1.8) in the form

F11[a;2a;   x]F11[b;2b;   -x]=F23[12(a+b),12(a+b+1);a+12,b+12,a+b;   x24].

Rathie and Choi [14] derived the Bailey identity (1.10) by using the same technique given in [13] and obtained four results closely related to it.

In fact, a number of identities associated with the product of generalized hypergeometric series have been investigated (see, e.g., [2, Chapter X]; see also [11, Entires 8.4.45–8.4.49]). In this paper, we aim to establish an identity involving the product of generalized hypergeometric series 2F2 by using the generalized Kummer-type II transformation (1.7) and the following identity due to Bailey [1] (see also [6, Exercise 7.24 (a)])

F01[-;a;   x]F01[-;b;   x]=F23[12(a+b),12(a+b-1);a,b,a+b-1;   4x].

The result presented here, being general, can be reduced to a number of relatively simple identities involving the product of generalized hypergeometric series, some of which are explicitly indicated to correspond to known ones.

2. Product Formula for 2F2

We present an identity involving product of 2F2, asserted in the following theorem.

Theorem 2.1

The following identity holds.

F22[a,1+d;2a+1,d;   x]F22[b,1+e;2b+1,e;   x]=ex{F23[12(a+b),12(a+b+1);a+12,b+12,a+b;   x24]+x(2a-d)2d(2a+1)   F23[12(a+b+1),12(a+b+2);a+32,b+12,a+b+1;   x24]+x(2b-e)2e(2b+1)   F23[12(a+b+1),12(a+b+2);a+12,b+32,a+b+1;   x24]+x2(2a-d)(2b-e)4de(2a+1)(2b+1)   F23[12(a+b+2),12(a+b+3);a+32,b+32,a+b+2;   x24]}.
Proof

Let

L:=e-xF22[a,1+d;2a+1,d;   x]F22[b,1+e;2b+1,e;   x].

Then

L:={e-x2F22[a,1+d;2a+1,d;   x]}{e-x2F22[b,1+e;2b+1,e;   x]}.

Using (1.7) in each factor of (2.2), we get

L={F01[-;a+12;   x216]+x(2a-d)2d(2a+1)F01[-;a+32;   x216]}×{F01[-;b+12;   x216]+x(2b-e)2e(2b+1)F01[-;b+32;   x216]}.

Expanding the right member of (2.3), we obtain

L=F01[-;a+1/2;   x216]F01[-;b+1/2;   x216]+x(2a-d)2d(2a+1)   F01[-;a+3/2;   x216]F01[-;b+1/2;   x216]+x(2b-e)2e(2b+1)   F01[-;a+1/2;   x216]F01[-;b+3/2;   x216]+x2(2a-d)(2b-e)4de(2a+1)(2b+1)   F01[-;a+3/2;   x216]F01[-;b+3/2;   x216].

Finally, using (1.11) in each term of the right member of the last identity, we are led to the right member of (2.1) with the factor ex deleted. This completes the proof.

3. Special Cases

The result (2.1), being general, can be reduced to yield a number of relatively simple identities, several of which are considered here.

  • Setting d = 2a in (2.1), we get e-x   F11[a;2a;x]F22[b,1+e;2b+1,e;   x]=F23[12(a+b),12(a+b+1);a+12,b+12,a+b;   x24]+x(2b-e)2e(2b+1)   F23[12(a+b+1),12(a+b+2);a+12,b+32,a+b+1;   x24].

    The identity (3.1) is found to be equivalent to a very recent result due to Kim et al. [5] who used a different method.

  • Taking d = 2a and e = 2b in (2.1), we have e-x   F11[a;2a;   x]F11[b;2b;   x]=F23[12(a+b),12(a+b+1);a+12,b+12,a+b;   x24].

    Using (1.3) in (3.2), we obtain Bailey’s identity (1.10). So the identity (2.1) may be regarded as a generalization of Bailey’s identity (1.10).

  • Setting e = d and b = a in (2.1), we find e-x{F22[a,1+d;2a+1,d;   x]}2=F12[a;a+12,2a;   x24]+x(2a-d)d(2a+1)   F12[(a+1);a+32,2a+1;   x24]+x2(2a-d)24d2(2a+1)2   F12[(a+1);a+32,2b+2;   x24].

    Taking d = 2a in (3.3) yields Preece’s identity (1.9). Therefore, the identity (3.3) can be looked upon as a generalization of Preece’s identity (1.9).

Acknowledgements

The authors would like to express their deep-felt thanks for the reviewer’s detailed and helpful comment to improve this paper as it stands.

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