Applications of the operator $H(\alpha,\beta)$ to the Humbert double hypergeometric functions

TL;DR

This paper develops new operational techniques using inverse symbolic operators to derive 34 decomposition formulas for Humbert hypergeometric functions, enhancing understanding of their structure and integral representations.

Contribution

It introduces novel inverse symbolic operator methods to systematically decompose Humbert hypergeometric functions, resulting in numerous new formulas and integral representations.

Findings

34 new decomposition formulas for Humbert functions

Operational representations based on inverse symbolic operators

Euler type integrals linked to Humbert functions

Abstract

By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Humbert hypergeometric functions $\Phi_1 $, $\Phi_2 $, $\Phi_3 $, $\Psi_1 $, $\Psi_2 $, $\Xi_1 $ and $\Xi_2 $. These operational representations are constructed and applied in order to derive the corresponding decomposition formulas. With the help of these inverse pairs of symbolic operators, a total 34 decomposition formulas are found. Euler type integrals, which are connected with Humbert's functions are found.