Solutions ofk-Hypergeometric Differential Equations

Abstract

We solve the second-order linear differential equation called thek-hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

“…In this section, we briefly review some basic definitions and facts concerning the k-hypergeometric series and the ordinary differential equation. Some surveys and literature for k-hypergeometric series and the k-hypergeometric differential equation can be found in Díaz et al [30,31], Krasniqi [32,33], and Mubeen et al [38,39].…”

“…It is clear that the k-hypergeometric series 2 F 1,k has evolved from the hypergeometric series 2 F 1 . Hence, we mention the works of Díaz et al [30,31], Krasniqi [32,33], Kokologiannaki [34], Mubeen et al (see [35][36][37][38][39][40]), Rehman et al [41,42], and the references therein for results on k-hypergeometric series and the homogeneous k-hypergeometric differential equation. In 2005, the Pochhanner k-symbol was developed by Díaz et al [30].…”

“…Mubeen [37] did not introduce the second-order linear k-hypergeometric differential equation defined by Equation (3) until 2013. Furthermore, in 2014, Mubeen et al [38,39] solved the k-hypergeometric differential equation by using the Frobenius method and gave its solution in the form of the so-called k-hypergeometric series 2 F 1,k introduced by Díaz et al [30]. In the case of k ∈ R + and f (z) = 0, the research for this question is very limited.…”

Certain inequalities involving the k-Struve function

“…For various properties of the k-gamma function and its applications to generalize other related functions such as k-beta function and k-digamma function, we refer the interested reader, for example, to [1–3] and the references cited therein.…”

On Generalized k-Fractional Derivative Operator

“…The solution of some integral equations involving confluent k-hypergeometric functions and k-analogue of Kummer's first formula are given in [12,13]. While the k-hypergeometric and confluent k-hypergeometric differential equations are introduced in [10].…”