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+1Fr(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? de F?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: 3F2 (?27z/4(1?z)3) and a reduction formula for the Srivastava-Daoust double hypergeometric function with the arguments (z,?z/4 ) are also derived. Many of the reduction formulas, which are established in this paper, are verified by using the software program, Mathematica. Some potential directions for further researches along the lines of this paper are also indicated.