Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle

TL;DR

This paper introduces new elliptic solutions to the $QD$-algorithm and discrete-time Toda chain, leading to novel elliptic hypergeometric orthogonal and biorthogonal polynomials with a dense spectrum on the unit circle.

Contribution

It constructs explicit elliptic solutions for the $QD$-algorithm and Toda chain, and derives new elliptic hypergeometric orthogonal polynomials with a dense spectrum.

Findings

New elliptic solutions of the $QD$-algorithm and Toda chain.

Explicit elliptic hypergeometric orthogonal and biorthogonal polynomials.

Degenerate case yields Krall-Jacobi polynomials.

Abstract

Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs.