Applications of an operator $H({\alpha ,\beta})$ to the Lauricella multivariable hypergeometric functions

TL;DR

This paper develops new symbolic operator techniques to derive 20 decomposition formulas and integral representations for Lauricella multivariable hypergeometric functions, enhancing understanding of their structure and relationships.

Contribution

It introduces novel inverse symbolic operator pairs to obtain decomposition formulas for Lauricella functions in multiple variables.

Findings

20 new decomposition formulas derived

Integral representations established for Lauricella functions

Operational methods applied to three-variable cases

Abstract

By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Lauricella's hypergeometric functions $F_A^{(r)}, F_B^{(r)}, F_C^{(r)}$ and $F_D^{(r)}$ in $r$ variables. In the three-variable case some of these operational representations are constructed and applied in order to derive the corresponding decomposition formulas when $r = 3$ . With the help of these new inverse pairs of symbolic operators, a total 20 decomposition formulas and integral representations are found.