On Bibasic Humbert hypergeometric function $\Phi_1$

TL;DR

This paper develops new $q$-recurrence, $q$-partial derivative, and summation formulas for the bibasic Humbert hypergeometric function $\

Contribution

It introduces novel $q$-calculus based relations and formulas for the bibasic Humbert hypergeometric function $\

Findings

Derived $q$-recurrence relations for $\

Established $q$-partial derivative relations for $\

Presented new summation formulas for $\

Abstract

The main aim of this work is to derive the $q$-recurrence relations, $q$-partial derivative relations and summation formula of bibasic Humbert hypergeometric function $\Phi_1$ on two independent bases $q$ and $q_{1}$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus.