On the recursion formulas of Lauricella matrix functions

The theory of matrix special functions has attracted considerable attention in the last two decades. Special matrix functions appear in the literature related to Statistics [4], Lie theory [9] and in connection with the second order matrix differential equations satisfied by matrix polynomials as well as matrix functions, for more detail see [5]–[8], [11]–[15]. Recursion formulas for the Appell functions have been studied in the literature, see Saad and Srivastava [16] and Wang [23]. Authors carried out a systematic study of recursion formulas for the multivariable hypergeometric functions, [17]-[21]. Abd-Elmageed et. al. [1] have obtained numerous contiguous and recursion formulas satisfied by the first Appell matrix function, namely F1. Recently, Sahai and Verma obtain recursion formulas for one and two variable hypergeometric matrix functions in [18]. In the present paper, we study recursion formulas for Lauricella matrix function of three variables as well as k-variables. The paper is organized as follows. In Section 2, we give a brief review of basic definitions that are needed in the sequel. In Section 3, we obtain the recursion formulas for generalized Lauricella matrix function of k-variables. Finally, in Section 4, the recursion formulas for three variable Lauricella matrix functions are given.