An extension of basic Humbert hypergeometric functions Φ1 , Φ2 and Φ3

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 several interesting $q$-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 hypergeometric functions $\mathbf{\Phi}_{1}$, $\mathbf{\Phi}_{2}$ and $\mathbf{\Phi}_{3}$ under constraints of symmetry parameters. These interesting properties, as special cases, include many known expansions of basic Humbert hypergeometric functions $\mathbf{\Phi}_{1}$, $\mathbf{\Phi}_{2}$ and $\mathbf{\Phi}_{3}$, and are also include particular interest in the area.