The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

Abstract

In this paper, we first introduce the incomplete beta matrix function ${\mathbb{B}}^z(M,K),z\in \mathbb{C}$, for parameter matrices M and K in ${\mathbb{C}}^{m\times m}$ using the neutrix limit and evaluation methods on their the partial derivatives. Also, we define a new extension of the Pochhammer matrix symbol in terms of the incomplete beta matrix functions. Furthermore, we then apply this extended matrix symbol for the incomplete first Appell hypergeometric matrix functions and obtain some of their properties.

Keywords:

• Incomplete beta matrix function
• incomplete Pochhammer matrix symbol
• incomplete Appell hypergeometric matrix function

2010 Mathematics Subject Classifications:

• 15A15
• 33D15
• 33C05
• 33C45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Larg Groups [grant number R.G.P.2/25/44], and the National Natural Science Foundation of China (11971201, 11601525).