Transformations and Reductions of Srivastava-Daoust Type Double Hypergeometric Functions

The generalized hypergeometric functions of one, two and more variables and allied Special Functions, and their associated transformations, reductions and summations are potentially useful, not only as solutions of ordinary and partial differential equations, but also in the widespread problems in the mathematical, physical, engineering, and statistical sciences. In the same context, by applying two well-known Euler’s transformations for the Gauss hypergeometric function, Liu and Wang, in 2014, established five general double series transformations involving some appropriately bounded sequences of complex numbers, and used the derived results to deduce many additional transformations, reductions and summations for the Kampé de Fériet function. The main objective of this work is to provide an essential and convenient methodology to prove, the five general transformations due to Liu and Wang, by applying the classical hypergeometric summation theorems. Motivated from the developed methodology, numerous additional general double series transformations involving some appropriately bounded sequences of complex numbers, are investigated. It is also shown that the newly obtained transformations, not only contain the five general transformations due to Liu and Wang but also lead to many other additional transformations and reductions for the Kampé de Fériet and the Srivastava-Daoust type double hypergeometric series. Further special cases are also examined.