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 Mat. Sb., 1993, Volume 184, Number 11, Pages 63–92 (Mi msb1026)

Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients

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

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

Abstract: Under very general conditions on the complex coefficients of a three-term recurrence relation, it is proved that 'almost all' zeros of the polynomials generated by these relations 'accumulate' on a certain segment in the complex plane. From this result follow the convergence of diagonal Padé approximants and a generalization of Van Vleck's theorem on the convergence of $S$-fractions. Another interesting application is an extension of the so-called Nevai–Blumenthal class of polynomials $M(a,2b)$ to the case when $a,b\in{\mathbb C}$.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 80:2, 309–333

Bibliographic databases:

UDC: 517.5
MSC: Primary 30E10; Secondary 42C05

Citation: D. Barrios, G. L. Lopes, E. Torrano, “Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients”, Mat. Sb., 184:11 (1993), 63–92; Russian Acad. Sci. Sb. Math., 80:2 (1995), 309–333

Citation in format AMSBIB
\Bibitem{BarLopTor93} \by D.~Barrios, G.~L.~Lopes, E.~Torrano \paper Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients \jour Mat. Sb. \yr 1993 \vol 184 \issue 11 \pages 63--92 \mathnet{http://mi.mathnet.ru/msb1026} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1251002} \zmath{https://zbmath.org/?q=an:0827.42014} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 80 \issue 2 \pages 309--333 \crossref{https://doi.org/10.1070/SM1995v080n02ABEH003527} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995QR47400004} 

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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. D. Barrios, G. L. Lopes, E. Torrano, “Polynomials generated by a three-term recurrence relation with asymptotically periodic complex coefficients”, Sb. Math., 186:5 (1995), 629–659
2. Lopez G. Marcellan F. Vanassche W., “Relative Asymptotics for Polynomials Orthogonal with Respect to a Discrete Sobolev Inner-Product”, Constr. Approx., 11:1 (1995), 107–137
3. A.Almendral Vázquez, “The Spectrum of a Periodic Complex Jacobi Matrix Revisited”, Journal of Approximation Theory, 105:2 (2000), 344
4. Bernhard Beckermann, “Complex Jacobi matrices”, Journal of Computational and Applied Mathematics, 127:1-2 (2001), 17
5. M.J. Cantero, L. Moral, L. Velázquez, “Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle”, Linear Algebra and its Applications, 362 (2003), 29
6. M.J. Cantero, L. Moral, L. Velázquez, “Measures on the unit circle and unitary truncations of unitary operators”, Journal of Approximation Theory, 139:1-2 (2006), 430
7. Borcea J., Bogvad R., Shapiro B., “On Rational Approximation of Algebraic Functions”, Adv. Math., 204:2 (2006), 448–480
8. Barrios Rolania D., Lopez Lagomasino G., “Asymptotic Behavior of Solutions of General Three Term Recurrence Relations”, Adv. Comput. Math., 26:1-3 (2007), 9–37
9. de la Calle Ysern B., “A Walk through Approximation Theory”, Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemporary Mathematics, 507, 2010, 25–86
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