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

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.