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

In this paper, we first introduce the incomplete beta matrix function B-z(M, K), Z is an element of C, for parameter matrices M and K in C-mxm 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.