Certain new formulas for bibasic Humbert hypergeometric functions ${\Psi }_1$ and ${\Psi }_2$

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.