Preprint Article Version 1 This version is not peer-reviewed

# A New Extension of Beta and Hypergeometric Functions

Version 1 : Received: 5 January 2018 / Approved: 9 January 2018 / Online: 9 January 2018 (07:08:48 CET)

How to cite: Rahman, G.; Kanwal, G.; Nisar, K.S.; Ghaffar, .A. A New Extension of Beta and Hypergeometric Functions. Preprints 2018, 2018010074 (doi: 10.20944/preprints201801.0074.v1). Rahman, G.; Kanwal, G.; Nisar, K.S.; Ghaffar, .A. A New Extension of Beta and Hypergeometric Functions. Preprints 2018, 2018010074 (doi: 10.20944/preprints201801.0074.v1).

## Abstract

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

## Subject Areas

beta function; extended beta function; hypergeometric function; extended hypergeometric function; confluent hypergeometric function; extended confluent hypergeometric function; Mellin transform; beta distribution; mean; variance; transformation formula; summation formula