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Mosc. Math. J., 2014, Volume 14, Number 1, Pages 1–27 (Mi mmj512)  

This article is cited in 3 scientific papers (total in 3 papers)

Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations

Philippe Biane

CNRS, IGM, Université Paris-Est, Champs-sur-Marne, France

Abstract: We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlevé equations, in a Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.

Key words and phrases: orthogonal polynomials Painlevé equations scattering theory.

DOI: https://doi.org/10.17323/1609-4514-2014-14-1-1-27

Full text: http://www.mathjournals.org/.../2014-014-001-001.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 33E17, 34L25, 39A45, 42C05
Received: July 6, 2010; in revised form June 18, 2013
Language:

Citation: Philippe Biane, “Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations”, Mosc. Math. J., 14:1 (2014), 1–27

Citation in format AMSBIB
\Bibitem{Bia14}
\by Philippe~Biane
\paper Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlev\'e equations
\jour Mosc. Math.~J.
\yr 2014
\vol 14
\issue 1
\pages 1--27
\mathnet{http://mi.mathnet.ru/mmj512}
\crossref{https://doi.org/10.17323/1609-4514-2014-14-1-1-27}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3221944}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000342789200001}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. S. Witte, “Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy”, Nagoya Math. J., 219 (2015), 127–234  crossref  mathscinet  zmath  isi  elib  scopus
    2. N. Joshi, N. Nakazono, Y. Shi, “Lattice equations arising from discrete Painlevé systems: II. $A^{(1)}_4$ case”, J. Phys. A, 49:49 (2016), 495201, 39 pp.  crossref  mathscinet  zmath  isi  scopus
    3. N. Joshi, N. Nakazono, “Lax pairs of discrete Painlevé equations: $(A_2+A_1)^{(1)}$ case”, Proc. A, 472:2196 (2016), 20160696, 14 pp.  crossref  mathscinet  zmath  isi  scopus
  • Moscow Mathematical Journal
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