An extension of basic Humbert hypergeometric functions {\Phi}1, {\Phi}2 and {\Phi}3

TL;DR

This paper systematically studies extended basic Humbert hypergeometric functions within $q$-calculus, deriving various formulas, relations, and representations that unify and generalize known results in the field.

Contribution

It introduces new extended forms of basic Humbert hypergeometric functions and explores their properties, including derivative, relation, recurrence, differential equations, and integral representations.

Findings

Derived $q$-partial derivative formulas

Established $q$-contiguous and recurrence relations

Presented new $q$-integral representations

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.