Certain Integral Formulae Associated with the Product of Generalized Hypergeometric Series and Several Elementary Functions Derived from Formulas for the Beta Function

Abstract

The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the generalized hypergeometric series p+1Fp and the integrands of the three Beta function formulae. Among the many particular instances for our formulae, several are stated clearly. Moreover, an intriguing inequality that emerges throughout the proving procedure is shown. It is worth noting that the three integral formulae shown here may be expanded further by using a variety of more generalized special functions than p+1Fp. Symmetry occurs naturally in the Beta and p+1Fp functions, which are two of the most important functions discussed in this study.

Keywords

• gamma function
• beta function
• generalized hypergeometric functions <i>_pF_q</i>
• summation formulas for <i>_pF_q</i>