Certain new formulas for bibasic Humbert hypergeometric functions $\Psi_{1}$ and $\Psi_{2}$

TL;DR

This paper develops new $q$-analogues of recurrence, derivative, and integral formulas for bibasic Humbert hypergeometric functions $\\Psi_{1}$ and $\\Psi_{2}$, expanding their theoretical framework using $q$-calculus.

Contribution

It introduces novel $q$-recurrence, transformation, and summation formulas for bibasic Humbert hypergeometric functions on two bases, with new developments and special case identities.

Findings

Derived new $q$-recurrence relations for $\\Psi_{1}$ and $\\Psi_{2}$.

Established transformation and summation formulas in the bibasic setting.

Identified special cases when the two bases are equal, leading to simplified identities.

Abstract

The main aim of the present work is to give some interesting the $q$-analogues of various $q$-recurrence relations, $q$-recursion formulas, $q$-partial derivative relations, $q$-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions $\Psi_{1}$ and $\Psi_{2}$ on two independent bases $q$ and $p$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types $\Psi_{1}$ and $\Psi_{2}$ are established when the two independent bases $q$ and $p$ are equal.