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Unveiling new perspectives of hypergeometric functions using umbral techniques

Giuseppe Dattoli, Mehnaz Haneef, Subuhi Khan, Silvia Licciardi

Abstract

The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of Gauss hypergeometric functions and extend the formalism to certain generalized forms of these functions. It is shown that suggested approach is particularly efficient for evaluating integrals involving hypergeometric functions and their combination with other special functions.

Unveiling new perspectives of hypergeometric functions using umbral techniques

Abstract

The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of Gauss hypergeometric functions and extend the formalism to certain generalized forms of these functions. It is shown that suggested approach is particularly efficient for evaluating integrals involving hypergeometric functions and their combination with other special functions.
Paper Structure (5 sections, 3 theorems, 117 equations)

This paper contains 5 sections, 3 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Integrals involving hypergeometric and Gaussian functions
  3. Applications
  4. Umbral methods and hypergeometric differential equations
  5. Concluding remarks

Key Result

Theorem 1

For the function ${_1{F}_2}[a;b,c;\beta x]$, the following integral representation holds:

Theorems & Definitions (9)

  • Remark 1
  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • proof
  • Theorem 2
  • Example 4
  • Corollary 1