2013/03/31 by Paweł J. Szabłowski
Computer Science · Mathematics · #Algebra over a field #Combinatorics #Discrete orthogonal polynomials #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Mathematics #Matrix Theory and Algorithms #Orthogonal polynomials #Pure mathematics #math.AP #msc:33D45 #msc:42C05 #msc:42C10 #msc:60J35
paper · pdf · doi:10.1016/j.amc.2014.11.112
published as Applied Mathematics and Computation, 252 (2015) pp. 215-228 · 18 pages
arxiv created 2014/12/05 · openalex publication_date 2014/12/26 · arxiv updated 2014/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials \ pn\ n≥ 0 that are orthogonal with respect to this distribution, coefficients of expansion of xn in the series of pj, j≤ n, two sequences of coefficients of the 3-term recurrence of the family of \ pn\ n≥ 0, the so called "linearization coefficients" i.e. coefficients of expansion of % pnpm in the series of pj, j≤ m+n.\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials \ pn\ n≥ 0, we express with their help: coefficients of the power series expansion of pn, coefficients of expansion of xn in the series of pj, j≤ n, moments of the distribution that makes polynomials \ pn\ n≥ 0 orthogonal. \newline Further having two different families of orthogonal polynomials \ pn\ n≥ 0 and \ qn\ n≥ 0 and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of pn in the series of qj, j≤ n.\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials \ pj( x)) \ j=0n as a linear transformation of the vector \ xj\ j=0n by some lower triangular (n+1)× (n+1) matrix Πn.