Some Generalized Special Functions and their Properties

Open Access

Original research article

Author(s):

Shahid Mubeen^a, Syed Ali Haider Shah^a, Gauhar Rahman^b, Kottakkaran Sooppy Nisar^c, Thabet Abdeljawad^d

^a Department of Mathematics, University of Sargodha, Sargodha, Pakistan.
^b Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Pakistan.
^c Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia.
^d Department of Mathematics, Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.

Advances in the Theory of Nonlinear Analysis and its Applications 6 (1), 45-65.

Received: July 14, 2020

Accepted: October 9, 2021

Published: October 16, 2021

Abstract

In this present paper, first, we investigate a new generalized Pochhammer’s symbol and its various properties in terms of a new symbol (s;k), where s;k>0. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of (s;k). Also, we define a 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 and 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.

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APA Style

Mubeen, S., Shah, S., Rahman, G., Nisar, K., & Abdeljawad, T. (2022). Some Generalized Special Functions and their Properties. Advances in the Theory of Nonlinear Analysis and its Application , 6 (1), 45-65.