Some properties of Ψ-gamma, Ψ-beta and Ψ-hypergeometric matrix functions

Abstract

In this paper, we investigate the matrix analogues of the Ψ-beta and Ψ-gamma functions, as well as their properties. With the help of the Ψ-beta matrix function (BMF), we introduce the Ψ-Gauss hypergeometric matrix function (GHMF) and the Ψ-Kummer hypergeometric matrix function (KHMF) and derive certain properties for these matrix functions. Finally, the Ψ-Appell and the Ψ-Lauricella matrix functions are defined by applications of the Ψ-BMF, and their integral representations are also given.

Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

“…[29]). Traditional beta distributions were introduced in [11][12][13]28,30] using extended beta functions. They suggested that these distributions could help analyze and review techniques employed in specific circumstances during project evaluation and review.…”

“…Remark 6. The incomplete generalized beta-logarithmic matrix function in (55) can be reduced to numerous simple incomplete extended beta matrix functions (see, e.g., [4,13]).…”

“…On the other hand, recent extensions of special functions build upon the work of esteemed researchers such as Abdalla et al [4,5], Abd-Elmageed et al [6], Hidan et al [7], Fuli He et al [8], and Cuchta et al [9], who have shown a strong interest in studying the extension of special functions in matrix arguments. Contributions to the field by considering various extensions of the gamma, beta, and hypergeometric matrix functions have been documented in [10][11][12][13][14]. Inspired by earlier studies, a new extension of the beta function in its matrix version is presented: the extended beta-logarithmic matrix function (EBLMF).…”