The growth of laguerre matrix polynomials on bounded intervals

Abstract Let A be a matrix in C r×r such that Re (z) > −1 2 for all the eigenvalues of A and let {π n ( A , 1 2 ) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π n (A, 1 2 ) (x) = O(n α(A)/2 ln r−1 (n)) and π n+1 (A, 1 2 ) (x) − π n (A, 1 2 ) (x) = O(n (α(A)−1)/2 ln r−1 (n)) uniformly on bounded intervals, where α(A) = max{Re(z); z eigenvalue of A}.