A few remarks on orthogonal polynomials

TL;DR

This paper develops methods to connect moments, polynomial expansions, and recurrence coefficients of orthogonal polynomials, enabling efficient computation and interrelation of these properties for a given distribution or polynomial family.

Contribution

It introduces a novel approach using linear transformations to relate moments, polynomial expansions, and recurrence coefficients of orthogonal polynomials, and extends to connections between different polynomial families.

Findings

Efficient computation of polynomial coefficients from moments.

Explicit formulas for moments from recurrence coefficients.

Method to find connection coefficients between polynomial families.

Abstract

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $% p_{n}p_{m}$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ 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 $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{\Pi }_{n}.$