A note on an expansion of hypergeometric functions of two variables

Abstract

1. In 1927 Fox [5] obtained the expansions of a product of two Bessel functions in a series of product of a Bessel function and a Gaussian hypergeometric function. Later on Rice [8] and Bailey [2] discovered a number of results of this type, some of them contained the result of Fox as a particular case. Recently, Srivastava [9] gave four general expansions of products of generalised hypergeometric functions in a series of product of generalised hypergeometric functions of two variables and a Gaussian hypergeometric function, which incorporated as a special case the results of Fox, Rice and Bailey mentioned above. The aim of the present note is to derive an expansion of a generalised hypergeometric function of two variables in a series of product of generalised hypergeometric functions of two variables and a generalised hypergeometric function. The result deduced is further generalised to an expansion of a Meijer's G-function of two variables (defined recently by Agarwal) in a series of product of a Meijer's G-function and a hypergeometric function of two variables. The results obtained are very general and contain as special cases the expansions of Jerry L. Fields and Jet Wimp [6], L. Carlitz and W. Alsalam [4], Meijer [7] and many other results. The results are obtained by the use of the Laplace transform and the inverse Laplace transform as has been done by the author elsewhere also [11], [12]. The following notation due to Chaundy [3] shall be used to represent the hypergeometric function of higher order and of two variables