Some Generalized Special Functions and their Properties

Abstract

In the present paper, first, we investigate a generalized Pochhammer's symbol and its various properties in terms of a new symbol (s, k), where s, k > 0. Then, we define a generalization of gamma and beta functions and their various associated properties in the form of (s, k). Also, we define new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of (s, k). We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results, properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

Keywords

• pochhammer symbol
• beta function
• gamma function
• generalized pochhammer symbol
• generalized hypergeometric function