On the matrix version of extended Bessel functions and its application to matrix differential equations

In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions. Finally, as an application, we study a kind of second-order matrix differential equations. We prove that the extended modified Bessel matrix function is a particular solution to this kind of differential equations.