On hypergeometric functions and Pochhammer k-symbol

We introduce the k-generalized gamma function Γₖ, beta function Bₖ, and Pochhammer k-symbol (x) ₍, ₊. We prove several identities generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. We provided integral representation for the Γₖ and Bₖ functions.