New Extension of Beta Function and Its Applications

Abstract

In the present paper, new type of extension of classical beta function is introduced and its convergence is proved. Further it is used to introduce the extension of Gauss hypergeometric function and confluent hypergeometric functions. Then we study their properties, integral representation, certain fractional derivatives, and fractional integral formulas and application of these functions.

Some Inequalities for an Extended Beta Function

“…Other representations of this extended beta function and its connections with some other special functions are discussed in [3] and [8]. The importance of this last type of functions is highlighted in [4] by some of their applications. It is worth mentioning to pay attention that the extension of the beta function presented in the paper [3] is different from that given in [4].…”

“…The importance of this last type of functions is highlighted in [4] by some of their applications. It is worth mentioning to pay attention that the extension of the beta function presented in the paper [3] is different from that given in [4].…”

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.…”

A New Range of Modified Gamma and Beta Functions

“…Recently, many generalizations, modifications, extensions and variants of gamma and beta functions [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] have been proposed.…”