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


SIGMA 3 (2007), 003, 18 pages      math.CA/0701135      http://dx.doi.org/10.3842/SIGMA.2007.003
Contribution to the Vadim Kuznetsov Memorial Issue

Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction

Luc Vinet a and Alexei Zhedanov b
a) Université de Montréal, PO Box 6128, Station Centre-ville, Montréal QC H3C 3J7, Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received October 07, 2006, in final form December 12, 2006; Published online January 04, 2007

Abstract
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.

Key words: Laurent biorthogonal polynomials; associated Legendre polynomials; elliptic integrals.

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