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 {}_3E_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.

Keywords

• elliptic Frobenius determinant
• QD-algorithm
• orthogonal and biorthogonal polynomials on the unit circle
• dense point spectrum
• elliptic hypergeometric functions
• Krall-Jacobi orthogonal polynomials
• quadratic operator pencils