Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 033, 30 pages      arXiv:0809.2574
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

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

Satoshi Tsujimoto a and Alexei Zhedanov b
a) Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received November 30, 2008, in final form March 15, 2009; Published online March 19, 2009

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 3E2(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.

Key words: 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.

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