On some formulas for the $k$-analogue of Appell functions and generating relations via $k$-fractional derivative

Abstract

Our present investigation is mainly based on the $k$-hypergeometric functions which are constructed by making use of the Pochhammer $k$-symbol \cite{Diaz} which are one of the vital generalization of hypergeometric functions. We introduce $k$-analogues of $F_{2}\ $and $F_{3}$ Appell functions denoted by the symbols $F_{2,k}\ $and $F_{3,k}\ $respectively, just like Mubeen et al. did for $F_{1}$ in 2015 \cite{Mubeen6}. Meanwhile, we prove some main properties namely integral representations, transformation formulas and some reduction formulas which help us to have relations between not only $k$-Appell functions but also $k$-hypergeometric functions. Finally, employing the theory of Riemann Liouville $k$-fractional derivative \cite{Rahman} and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for $k$-analogue of hypergeometric functions and Appell functions.