2011/02/08 by Jorge Borrego, Mirta Castro, Borrego, Jorge +4 · 1 citation
Computer Science · Mathematics · #42C05 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Polynomial and algebraic computation #math.CA #msc:42C05
paper · pdf · doi:10.48550/arxiv.1102.1578
17 pages
arxiv created 2011/02/08 · openalex publication_date 2011/02/08 · arxiv updated 2011/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a family of weight matrices W of the form T(t)T^*(t), T(t)=e^\mathscrAte^\mathscrDt2, where \mathscrA is certain nilpotent matrix and \mathscrD is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size N× N and depend on N parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials F2, F1 and F0 (independent of n) of degrees not bigger than 2, 1 and 0 respectively. For size 2× 2, we find an explicit expression for a sequence of orthonormal polynomials with respect to W. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.