On Bibasic Humbert hypergeometric function ${\mathrm{\Phi }}_1$

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 ${\mathrm{\Phi }}_1$ 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.