Some Formulas for New Quadruple Hypergeometric Functions

Jihad A Younis,Ashish Verma,Hassen Aydi,Ahmet Ocak Akdemir

doi:10.1155/2021/5596299

Abstract

In this paper, we aim to introduce six new quadruple hypergeometric functions. Then, we investigate certain formulas and representations for these functions such as symbolic formulas, differential formulas, and integral representations.

Highlights

Hypergeometric functions of several variables play an important role in diverse areas of science and engineering. e developments in applied mathematics, mathematical physics, chemistry, combinatorics, statistics, numerical analysis, and other areas have led to increasing interest in the study of multiple hypergeometric functions
Many authors have studied a number of formulas involving hypergeometric functions
In [7], Exton presented twenty-one complete hypergeometric functions in four variables denoted by symbols K1, K2, . . . , K21

Summary

Introduction

Hypergeometric functions of several variables play an important role in diverse areas of science and engineering. e developments in applied mathematics, mathematical physics, chemistry, combinatorics, statistics, numerical analysis, and other areas have led to increasing interest in the study of multiple hypergeometric functions. In [7], Exton presented twenty-one complete hypergeometric functions in four variables denoted by symbols K1, K2, . In [8], Sharma and Parihar defined eightythree complete quadruple hypergeometric functions, namely, F(14), F(24), . In [10], the authors discovered the existence of twenty additional complete hypergeometric functions in four variables X(314), X(324), . By using the conventions and notations above, we introduce further quadruple hypergeometric functions as follows: Journal of Mathematics. (a)m is the Pochhammer symbol defined (for a, m ∈ C), in terms of the familiar Gamma function Γ, by Appell’s double hypergeometric function F2 is defined as follows [13]: F2(a, b, c; d, e; x, y).

Symbolic Formulas

Integral Representations

Operator Formulas