A $(p,\nu )$-extension of the Appell function $F_1(\cdot )$ and its properties

In this paper, we obtain a $(p,v)$-extension of the Appell hypergeometric function $F_1(\cdot )$, together with its integral representation, by using the extended Beta function $B_{p,v}(x,y)$ introduced in arXiv:1502.06200. Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell function$F_{1,p,v}(\cdot )$ involving Meijer’s $G$-function are obtained.