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.

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关于扩展贝塞尔函数的矩阵版本及其在矩阵微分方程中的应用

在本文中，我们重点研究了贝塞尔矩阵函数的扩展和修正的贝塞尔矩阵函数。我们首先介绍了扩展Bessel矩阵函数和第一类利用扩展Beta矩阵函数的扩展修正Bessel矩阵函数。然后我们建立了这些函数的积分表示、微分公式和超几何表示。最后，作为一个应用，我们研究了一类二阶矩阵微分方程。我们证明了扩展修正贝塞尔矩阵函数是这类微分方程的特解。