New extension of beta, Gauss and confluent hypergeometric functions

Abstract

There are many extensions and generalizations of Gamma and Beta functions in the literature. However, a new extension of the extended Beta function , 1 2 ; 1 , 2 ( 1 , 2 ) was introduced and presented here because of its important properties. The new extended Beta function has symmetric property, integral representations, Mellin transform, inverse Mellin transform and statistical properties like Beta distribution, mean, variance, moment and cumulative distribution which ware also presented. Finally, the new extended Gauss and Confluent Hypergeometric functions with their propertied were introduced and presented.

The Beta‐Pochhammer and its application to arbitrary threshold phase error probability of a vector in Gaussian noise

“…The k ‐Pochhammer symbol was introduced by Diaz and Pariguan [8] based on the Gamma weighting 𝑡 𝑎−1 𝑒 −𝑡 with further generalizations by, for example, Rehman et al [9], Raissouli and El‐Soubhy [10], and Saboor et al [11]. Further, a wide range of extended Pochhammer symbols based on the Beta weighting 𝑡 𝑎−1 (1 − 𝑡) 𝑏−1 have been presented by authors such as Marfaing [12], Chand et al [13], Srivastava et al [14], Abubakar et al [15], Palsaniya et al [16], and Ghanim and Al‐Janaby [17]. These symbols offer a condensed mathematical structure that can lead to a consistent language that establishes connections across a plethora of concepts.…”