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

TL;DR

This paper introduces and studies $k$-analogues of Appell functions based on $k$-hypergeometric functions, deriving properties, relations, and generating formulas using $k$-fractional derivatives, expanding the theoretical framework of hypergeometric functions.

Contribution

The paper defines $k$-analogues of Appell functions and establishes their properties, relations, and generating formulas using $k$-fractional calculus, which is a novel extension in the field.

Findings

Derived integral representations and transformation formulas for $k$-Appell functions.

Established relations between $k$-hypergeometric and $k$-Appell functions.

Obtained linear and bilinear generating relations using $k$-fractional derivatives.

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.