Some Zero-Balanced Terminating Hypergeometric Series and Their Applications

Abstract

Various families of such Special Functions as the hypergeometric
functions of one, two and more variables, and their associated
summation, transformation and reduction formulas, 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.
The main object of this paper is first to establish four general
double-series identities, which involve some suitably-bounded
sequences of complex numbers, by using
zero-balanced terminating hypergeometric summation theorems
for the generalized hypergeometric series
$~{}_{r+1}F_r(1)\;\;(r=1,2,3)$ in conjunction with
the series rearrangement technique. The sum (or difference)
of two general double hypergeometric functions of the
Kamp\'{e} de F\'{e}riet type are then obtained in terms of
a generalized hypergeometric function under appropriate
convergence conditions. A closed form of the following
Clausen hypergeometric function:
$${}_3F_2\left(-\frac{27z}{4(1-z)^{3}}\right)$$
and a reduction formula for the Srivastava-Daoust double
hypergeometric function with the arguments $(z,-\frac{z}{4})$
are also derived. Many of the reduction formulas, which are
established in this paper, are verified by
using the software program, {\it Mathematica}. Some
potential directions for further researches along the lines
of this paper are also indicated.