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 Mat. Sb., 1995, Volume 186, Number 5, Pages 3–34 (Mi msb34)

Polynomials generated by a three-term recurrence relation with asymptotically periodic complex coefficients

D. Barriosa, G. L. Lopesb, E. Torranoa

a University of the Basque Country
b Carlos III University of Madrid

Abstract: In a previous paper of the authors the location of zeros of polynomials generated by a three-term recurrence relation with complex coefficients satisfying rather general conditions was studied. In particular, it was proved there that when these coefficients have limits in the complex plane, there are asymptotics of the ratio as in the Nevai–Blumenthal class of orthogonal polynomials. In this paper the case of asymptotically periodic recurrence coefficients is studied and the results known for the case of real recurrence coefficients are extended. Applications to rational approximation and continued fractions are presented.

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English version:
Sbornik: Mathematics, 1995, 186:5, 629–659

Bibliographic databases:

UDC: 517.5
MSC: 30E10, 30B70, 41A21, 42C05

Citation: D. Barrios, G. L. Lopes, E. Torrano, “Polynomials generated by a three-term recurrence relation with asymptotically periodic complex coefficients”, Mat. Sb., 186:5 (1995), 3–34; Sb. Math., 186:5 (1995), 629–659

Citation in format AMSBIB
\Bibitem{BarLopTor95} \by D.~Barrios, G.~L.~Lopes, E.~Torrano \paper Polynomials generated by a~three-term recurrence relation with asymptotically periodic complex coefficients \jour Mat. Sb. \yr 1995 \vol 186 \issue 5 \pages 3--34 \mathnet{http://mi.mathnet.ru/msb34} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1341082} \zmath{https://zbmath.org/?q=an:0855.41007} \transl \jour Sb. Math. \yr 1995 \vol 186 \issue 5 \pages 629--659 \crossref{https://doi.org/10.1070/SM1995v186n05ABEH000034} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TC19700001} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Bernhard Beckermann, “On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator”, Journal of Approximation Theory, 99:2 (1999), 369
2. Lubinsky, DS, “Asymptotics of orthogonal polynomials: Some old, some new, some identities”, Acta Applicandae Mathematicae, 61:1–3 (2000), 207
3. A.Almendral Vázquez, “The Spectrum of a Periodic Complex Jacobi Matrix Revisited”, Journal of Approximation Theory, 105:2 (2000), 344
4. Beckermann, B, “Complex Jacobi matrices”, Journal of Computational and Applied Mathematics, 127:1–2 (2001), 17
5. Peherstorfer, F, “Inverse images of polynomial mappings and polynomials orthogonal on them”, Journal of Computational and Applied Mathematics, 153:1–2 (2003), 371
6. Baratchart, L, “Multipoint Pade approximants to complex Cauchy transforms with polar singularities”, Journal of Approximation Theory, 156:2 (2009), 187
7. de la Calle Ysern B., “A Walk through Approximation Theory”, Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemporary Mathematics, 507, 2010, 25–86
8. V. V. Borzov, E. V. Damaskinsky, “The discrete spectrum of Jacobi matrix related to recurrence relations with periodic coefficients”, J. Math. Sci. (N. Y.), 213:5 (2016), 694–705
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