Elliptic solutions of the Toda chain and a generalization of the Stieltjes-Carlitz polynomials

TL;DR

This paper introduces new elliptic solutions to the restricted Toda chain, leading to a novel class of orthogonal polynomials that generalize Stieltjes-Carlitz polynomials, with explicit recurrence and weight functions.

Contribution

It constructs explicit elliptic solutions of the Toda chain and defines a new class of orthogonal polynomials extending classical elliptic polynomial families.

Findings

Explicit recurrence coefficients derived

Weight functions explicitly expressed

Degenerated cases recover known polynomial families

Abstract

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials. Relations between characteristic (i.e. positive definite) functions, Toda chain and orthogonal polynomials are developed in order to derive main properties of these polynomials. The recurrence coefficients and the weight function of these polynomials are expressed explicitly. In the degenerated cases of the elliptic functions the modified Meixner polynomials and the Krall-Laguerre polynomials appear.