An extension of basic Humbert hypergeometric functions Φ1, Φ2 and Φ3
Ayman Shehata
TL;DR
This work develops a comprehensive $q$-analytic extension of the Humbert hypergeometric functions $Phi1$, $Phi2$, $Phi3$ by introducing and exploiting $q$-difference operators, contiguity, recurrence and differential relations. It derives explicit $q$-derivative formulas for the functions with respect to $x$ and $y$, $q$-contiguous and $q$-recurrence relations, $q$-partial differential equations, as well as summation, transformation and $q$-integral representations. The results show that the $q$-extensions recover the classical Humbert functions in the limit $q o 1$. The framework provides a unified toolbox with potential applications in number theory, combinatorics, mathematical physics and engineering through transformed and integrated $q$-Humbert structures.
Abstract
Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain interesting several $q$-partial derivative formulas, $q$-contiguous function relations, $q$-recurrence relations, various $q$-partial differential equations, summation formulas, transformation formulas and $q$-integrals representations for basic Humbert confluent hypergeometric functions under what constraints of parameters. These interesting properties, as special cases, include many known expansions of basic Humbert hypergeometric functions, and are also include particular interest in the area.