@article{article_1335407, title={Some $k$-Horn hypergeometric functions and their properties}, journal={Journal of New Results in Science}, volume={12}, pages={97–107}, year={2023}, DOI={10.54187/jnrs.1335407}, author={Çatak, Caner and Şahin, Recep and Olgun, Ali and Yağcı, Oğuz}, keywords={$k$-Gamma function, $k$-Hypergeometric function, $k$-Gauss hypergeometric function, Recursion formula}, 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 $_{2}{F}_{1}^{k}$.}, number={2}, publisher={Tokat Gaziosmanpasa University}