### 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.

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