Discrete fractional solutions to the k‐hypergeometric differential equation

Abstract

In this study, the discrete fractional nabla calculus operator is used to investigate the k‐hypergeometric differential equation for both homogeneous and nonhomogeneous states. To solve the guiding equation, we implement certain classical transformations and also constrain the parameters needed to determine them valued. In order to achieve these results, some equipment like the Leibniz rule, the index law, the shift operator, and the power rule are set out in the frame of the discreet fractional calculus. We use all of these tools to the governing equation for homogeneous and nonhomogeneous situations. The major benefit of the fractional nabla operator is that it implements singular differential equations and converts them into fractional order equations. As a result, several new exact fractional solutions of the given equation are constructed.

Some results on (p, k)-extension of the hypergeometric functions

“…Definition 2.3. [16,26,27] Let k ∈ R + and s 1 , s 2 , η ∈ C and s 3 ∈ C \ Z − 0 , then k-Gauss hypergeometric function is defined in…”

“…In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al [17], Mubeen et al [18][19][20], Rahman et al [21], Chinra et al [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al [24], Li and Dong [25], Yilmaz et al [26] and Yilmazer and Ali [27].…”

Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

“…Definition 3 (see [16,26,28]) Let k ∈ R + and s 1 , s 2 , η ∈ C and s 3 ∈ C\Z − 0 ; then, k-Gauss hypergeometric function is defined in…”

“…e k-hypergeometric differential equation of second order is defined in [18,25,26,28] Mathematical Problems in Engineering…”

“…In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta, and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al [17], Mubeen et al [18][19][20], Rahman et al [21], Chinra et al [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al [24], Li and Dong [25], Yilmaz et al [26], Hidan et al [27], and Yilmazer and Ali [28].…”