Laguerre matrix polynomials and systems of second-order differential equations (Q1334840)

The usual discussions on Laguerre polynomials are extended to those on Laguerre matrix polynomials. The class of Laguerre matrix polynomials is introduced as an infinite series which are solutions of second-order matrix differential equations \(tX'' + \{A + I(1 - t \lambda)\}\) \(X' = CX = 0\) where \(t\) is a real variable, \(\lambda\) is a complex number, \(I\) is the \(m \times m\) identity matrix, \(A,C\) are \(m \times m\) matrices and \(X\) is a matrix valued function of \(t\). An explicit expression for the Laguerre matrix polynomials, a three term matrix recurrence relation, an extended Rodrigues formula and orthogonality property are obtained. The possibility of applications of the result in mechanics, physics, chemistry is just pointed out.