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

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.

References in corpus (4)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Essays on the theory of elliptic hypergeometric functions
• Faces of Relativistic Toda Chain
• The Lanczos Algorithm for extensive Many-Body Systems in the Thermodynamic Limit

Cited by in corpus (4)

• A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula
• Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
• An explicit example of polynomials orthogonal on the unit circle with a dense point spectrum generated by a geometric distribution
• Elliptic solutions of the Toda chain and a generalization of the Stieltjes-Carlitz polynomials