Orthogonality Property of the Discrete$q$-Hermite Matrix Polynomials

Abstract

In this paper, we prove that the solution of the autonomous q -difference system D q Y x = A Y q x with the initial condition Y 0 = Y 0 where A is a constant square complex matrix, D q is the Jackson q -derivative and 0 < q < 1 , is asymptotically stable if and only if ℜ λ < 0 for all λ ∈ σ A where σ A is the set of all eigenvalues of A (the spectrum of A ). This results are exploited to provide the orthogonality property of the discrete q -Hermite matrix polynomials.