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Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials
Alain Lascoux1,2 and Piotr Pragacz3
1CNRS, Institut Gaspard Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée Cedex France
Alain.Lascoux@univ-mlv.fr
2Center for Combinatorics, LMPC, Nankai University, Tianjin 300071, P.R. China
3Institute of Mathematics, Polish Academy of Ściences, Sniadeckich 8, 00-956 Warszawa Poland
P.Pragacz@impan.gov.pl
Annals of Combinatorics 9 (3) p.301-319 September, 2005
AMS Subject Classification: 05E05 05E35 12D99 15A54 33C47 42C05
Abstract:
We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the Euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this allows us to interpret their Bezoutian as the Christoffel- Darboux kernel for a finite family of orthogonal polynomials, arising from the Euclidean algorithm. We give orthogonality properties of remainders, and reproducing properties of Bezoutians.
Keywords: Euclidean algorithm, orthogonal polynomials, Bezoutians, Schur functions, Lagrange interpolation, continued fractions

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