On hypergeometric functions and Pochhammer $k$-symbol

TL;DR

This paper introduces generalized gamma, beta, and Pochhammer functions with new identities and integral representations, extending classical special functions to a broader mathematical framework.

Contribution

It presents novel definitions and identities for the $k$-generalized gamma, beta, and Pochhammer functions, expanding the theory of special functions.

Findings

Derived identities generalizing classical functions

Provided integral representations for $ ext{Gamma}_k$ and $B_k$ functions

Extended the mathematical framework of special functions

Abstract

We introduce the $k$-generalized gamma function $\Gamma_k$, beta function $B_k$, and Pochhammer $k$-symbol $(x)_{n,k}$. We prove several identities generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. We provided integral representation for the $\Gamma_k$ and $B_k$ functions.