On a new class of double integrals involving Gauss's ${}_{2}F_{1}$ hypergeometric function

Sungtae Jun

doi:10.5269/bspm.42422

Abstract

In this paper, one hundred interesting double integrals involving Gauss's hypergeometric function in the form of four general integrals (twenty five each) have been evaluated in terms of gamma function. More than two hundred special cases have also been given.

Highlights

In the same paper, Kim et al [4] have evaluated a large number of double integrals by employing classical summation theorems such as those of Watson, Dixon and Whipple for the series 3F2 of unit argument
The coefficients Ai,j, Bi,j and Ci,j are given in Tables 1, 2 and 3 at the end of this paper
(1) In (2.1), let b = −2n and change a to a + 2n or let b = −2n − 1 and change a to a + 2n + 1, where n is zero or a positive integer. In both the such cases, we notice that one of the two terms appearing on the right-hand side of (2.1) will vanish and we get fifty interesting and new special cases given in the following two corollaries

Summary

Introduction

Kim et al [4] have evaluated a large number of double integrals by employing classical summation theorems such as those of Watson, Dixon and Whipple for the series 3F2 of unit argument. Inspired by the double integrals (1.3) and (1.4), our aim, in this paper, is to evaluate the following double integrals xc−1yα+c−1(1 − x)α−1(1 − y)β−1(1 − xy)c+j−α−β (i) 1); xy dxdy xc−1yα+c−1(1 − x)α−1(1 − y)β−1(1 − xy)c+j−α−β (iii)

Results

Conclusion