Exploration of k-hypergeometric polynomials and their mathematical implications

In current study, we focus on a mathematical concept called the k-hypergeometric polynomials. These polynomials are constructed using a mathematical tool called the Pochhammer k-symbol, as intro-duced by R. Diaz, E. Pariguan, On hypergeometric functions and pochhammer k-symbol, Divulg. Mat. 15(2007), 179-192. We develop several theorems related to these k-hypergeometric polynomials. Using these theorems, we derive two important functions: a multilinear generating function and a multilateral generating function for k-hypergeometric polynomials. These functions play a crucial role in our analy-sis. Furthermore, extend our research to explore the concept of the k-fractional secondary driver. This extension is based on the properties of k-hypergeometric polynomials and another mathematical entity known as the beta k-function. To make these connections, we utilize the Riemann-Liouville k-fractional process, as described by G. Rahman, S. Nisar Mubeen, K. Sooppy, On generalized k-fractional derivative operator, AIMS Math. 5(2020), 1936-1945. This has allowed us to establish some novel results, which are analogous to well-known mathematical transformations like the Mellin transformation. Additionally, we explore the relationships between our findings and other mathematical functions, such as hypergeo-metric and Appell? k-functions. In the last section of our paper, we delve into the relationship between k-hypergeometric polynomials and two specific mathematical functions: We also provide an integral repre-sentation of k-hypergeometric polynomials. Overall, our research paper contributes to the understanding of k-hypergeometric polynomials and their connections to various mathematical functions and transformations.