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SIGMA 5 (2009), 033, 30 pages arXiv:0809.2574
https://doi.org/10.3842/SIGMA.2009.033
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
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
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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