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Laguerre matrix polynomials and systems of second-order differential equations. (English) Zbl 0821.34010

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.
Reviewer: M.Dutta (Calcutta)

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A30 Linear ordinary differential equations and systems
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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