Due to the great success of hypergeometric functions of one
variable, a number of hypergeometric functions of two or more variables
have been introduced and explored. The aim of this paper is to provide the
extensions and generalizations of Kummer's first summation theorem for
the higher-order hypergeometric series, where numeratorial and denominatorial
parameters differ by positive integers, in the form of
r+2Fr+1[a, b, {nr + ζr} ; 1 + a − b + m, {ζr} ; −1],
with suitable convergence conditions. Where ζr is set of complex or real
numbers, {nr} is set of positive integers and suitable restrictions on the
value of m.
1. INTRODUCTION AND PRELIMINARIES
The enormous popularity and broad usefulness of the hypergeometric function
2F1 and the generalized hypergeometric functions pFq (p, q ∈ N0) of
one variable have inspired and stimulated a large number of researchers to
introduce and investigate hypergeometric functions of two or more variables
(see, e.g., [3, 16, 7, 21]).