On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere

A relaxed factorization is used to obtain many of the properties obeyed by the confluent hypergeometric functions. Their implications on the analytical solutions of some interesting physical problems are also studied. It is quite remarkable that, although these properties appear frequently in solving the Schroedinger equation, it has been not clear the role they play in describing the physical systems. The main objective of this communication is precisely to throw some light ...

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q...

Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain special cases of the main results presented here are also pointed out for the extended Gauss' hypergeometric and confluent hypergeometric functions.

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values m...

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions...

The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases of the generalized hypergeometric function $_qF_p$. The Appell hypergeometric functions $F_q$, $q=1,2,3,4$ are product of two hypergeometric functions $_2F_1$ that appear in many areas of mathematical physics. Here, we are interested in the...

In this article three expansion formulas for a generalized hypergeometric function $_4F_3$ are derived, when its upper parameters differ by integers. Though the results are special cases of a general continuation formula for $_pF_q$, they are sufficiently general and unify a number of known results.

Very recently a new series representation of Humbert's double hypergeometric series $\Phi_3$ in series of Gauss's $_2F_1$ function was given by one of us. The aim of this short research note is to provide an alternative proof of the result. A few interesting special cases are also given.

We obtain addition formulas for $_{p}F_{p}$ and $_{p+1}F_{p}$ generalized hypergeometric functions with general parameters. These are utilized in conjunction with integral representations of these functions to derive Kummer- and Euler-type transformations that express $_{p}F_{p}\left(x\right)$ and $_{p+1}F_p\left(x\right)$ in the form of sums of $_{p}F_{p}\left(-x\right)$ and $_{p+1}F_p\left(-x\right)$ functions, respectively.

The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions of two and more variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities several decomposition formulas are found, which express the aforeme...