[.sub.p][F.sub.q] is the generalized

hypergeometric series defined by (see, Andrews, et al.

Gauss hypergeometric function and confluent hypergeometric function are special cases of the generalized

hypergeometric series [sub.p][F.sub.q](p, q [member of] N) defined as (see [8, p.73]) and [9, pp.

Karlsson, Multiple Gaussian

hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, NY, USA, 1985.

The quantum operators are widely used in mathematic fields such as

hypergeometric series, complex analysis, orthogonal polynomials, combinatorics, hypergeometric functions, and the calculus of variations.

of the generalized

hypergeometric series (the both sides of this equality are rational functions of [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2]), and the expression

Other examples achieved by termination of the

hypergeometric series expansions of the functions of the Heun class include the recently reported inverse square root [4], Lambert-W step [5], and Lambert-W singular [6] potentials.

Wenchang, "Inversion Techniques and Combinatorial Identities: Balanced

Hypergeometric Series," Rocky Mountain J.

This paper presents a new method for finding identities for

hypergeometric series, covering the Appell-Lauricella

hypergeometric series, the (Gauss)

hypergeometric series, and the generalized

hypergeometric series.

Rahman, Basic

Hypergeometric Series (with a Foreword by Richard Askey), Encyclopedia of Mathematics and Its Applications, 35, Cambridge University Press, Cambridge, New York, Port Chester, Melbourne and Sydney, 1990; Second edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, London and New York, 2004.

We now give the definition of the

hypergeometric series which will be used in obtaining some integrals.

This led to remarkable discovery of Heines formula for a q-hypergeometric function as a generalization of the

hypergeometric series and its connection to the Ramanujan product formula, relation between Eulers identities and the Jacobi Triple product identity in the 19th century.

Since the

hypergeometric series in (83) converges absolutely in D, it follows that F([beta], [gamma], [delta]; z) defines an analytic function in D and plays an important role in the theory of univalent functions.