The growth of laguerre matrix polynomials on bounded intervals

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Abstract

Let A be a matrix in Cr×r such that Re(z) > −12 for all the eigenvalues of A and let n(A,12) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that πn(A,12) (x) = O(nα(A)/2lnr−1(n)) and πn+1(A,12) (x) − πn(A,12) (x) = O(n(α(A)−1)/2lnr−1(n)) uniformly on bounded intervals, where α(A) = max{Re(z); z eigenvalue of A}.

Keywords

Laguerre matrix polynomials
Gamma function
Asymptotics

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This work has been supported by the D.G.I.C.Y.T. Grant PB96-1321-C02-02.