Some $k$-Horn hypergeometric functions and their properties

Year 2023, Volume: 12 Issue: 2, 97 - 107, 31.08.2023

Caner Çatak , Recep Şahin , Ali Olgun , Oğuz Yağcı

https://doi.org/10.54187/jnrs.1335407

Abstract

In the theory of special functions, the $k$-Pochhammer symbol is a generalization of the Pochhammer symbol. With the help of the $k$-Pochhammer symbol, we introduce and study a new generalization of the $k$-Horn hypergeometric functions such as, ${G}_{1}^{k}$, ${G}_{2}^{k}$ and ${G}_{3}^{k}$. Furthermore, several investigations have been carried out for some important recursion formulae for several one variable and two variables $k$-hypergeometric functions. In the light of these studies, we introduce some important recursion formulae for several newly generalized $k$-Horn hypergeometric functions. Finally, we present several relations between some $k$-Horn hypergeometric functions ${G}_{1}^{k}$, ${G}_{2}^{k}$ and ${G}_{3}^{k}$, and $k$-Gauss hypergeometric functions $_{2}{F}_{1}^{k}$.

Keywords

$k$-Gamma function, $k$-Hypergeometric function, $k$-Gauss hypergeometric function, Recursion formula

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