A New Extension of Humbert Matrix Function and Their Properties

Special matrix functions appear in the literature related to statistics [1-4] and more recently in connection with matrix analogues of Laguerre, Hermite and Legendre differential equations and the corresponding polynomial families [5-7]. The connection between the Humbert matrix function and modified Bessel matrix function has been established in [8,9]. In recent papers [10,11], we defined and studied the Humbert matrix functions. The Kummer’s confluent hypergeometric function belongs to an important class of special functions of the mathematical physics with a large number of applications in different branches of the quantum mechanics atomic physics, quantum theory, nuclear physics, quantum electronics, elasticity theory, acoustics, theory of oscillating strings, hydrodynamics, random walk theory, optics, wave theory, fiber optics, electromagnetic field theory, plasma physics, the theory of probability and the mathematical statistics, the pure and applied mathematics in [3,4,12-14]. Recently, an extension to the Kummer matrix function of complex variable is appeared in [15]. The first author has earlier studied the certain Kummer matrix function of two complex variables under certain differential and integral operators [16]. The primary goal of this paper is to consider a new system of matrix functions, namely the composite Humbert matrix function, Humbert Kummer matrix function and composite Humbert Kummer matrix function. The paper is organized as follows: Section 2 is define and study of the composite Humbert matrix function. The convergence and integral form is established. In Section 3 an operational relation between a Humbert matrix function and Kummer matrix function is given. Integral expressions of Humbert Kummer matrix functions are deduced. In Section 4 we defined and studied of the composite Humbert Kummer matrix functions. Throughout this paper 0 will denote the complex plane. A matrix is a positive stable matrix in D P N N C  if   > 0 Re  for all where   P      P  is the set of all eigenvalues of and its two-norm denoted by P