Abstract
The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.
Highlights
The study of functions of matrices is a very popular topic in the Matrix Analysis literature
The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that satisfy second order differential equations with coefficients independent of n (Duran and Grunbaum 2005)
We extend the family of q-Laguerre polynomials (5) of complex variables to q-Laguerre matrix polynomials by means of studying the solutions of the second order matrix q-difference equations x(Dq2Y )(x) + [A + I]q − x qA+2I (DqY )(qx) + [α]qCY = 0 (6)
Summary
The study of functions of matrices is a very popular topic in the Matrix Analysis literature. Let Ym and Yn are solutions of matrix q-difference equation (6) corresponding to αm and αn respectively, we get xAeq(−qx )Ym(qx)Yn(qx)dqx = 0, m = n where the q-integral is the inverse of q-derivative (3) defined as Lemma 7 Let A and C are matrices inCr×r such that q−n ∈ σ (C) for all n ∈ N0 and a ∈ C, the matrix functions
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