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

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

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

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

where

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

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

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

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

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

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)])

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.

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

Theorem 2.1

The following identity holds.

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 (3.1)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 (3.2)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 (3.3)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).

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.