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

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

2008-09-15

arxiv.org/abs/0809.2574v2

SIGMA 5 (2009), 033, 30 pages

Mathematics

Classical Analysis and ODEs

Scientific paper

10.3842/SIGMA.2009.033

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.

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