Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

From orthogonal polynomials on the unit circle to functional equations via generating functions


Authors: María José Cantero and Arieh Iserles
Journal: Trans. Amer. Math. Soc. 368 (2016), 4027-4063
MSC (2010): Primary 42C05, 33C05, 39A05
DOI: https://doi.org/10.1090/tran/6454
Published electronically: September 15, 2015
MathSciNet review: 3453364
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We explore orthogonal polynomials on the unit circle whose Schur parameters are $ \{c\alpha ^n\}_{n=1}^\infty $, where $ 0<\vert\alpha \vert,\vert c\vert<1$. Specifically, we derive two different generating functions. The first can be represented explicitly in terms of sums of a $ q$-hypergeometric type and used to derive explicitly the underlying orthogonal polynomials, while the second obeys a functional differential equation and can be used to determine the asymptotic behaviour of these polynomials. Extending these constructs to orthogonal polynomials of the second kind, we are able to construct the Carathéodory function and examine the underlying orthogonality measure.


References [Enhancements On Off] (What's this?)

  • [1] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
  • [2] George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719
  • [3] L. Ya. Geronimus, Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Authorized translation from the Russian, Consultants Bureau, New York, 1961. MR 0133643
  • [4] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • [5] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. 4 (1993), no. 1, 1–38. MR 1208418, https://doi.org/10.1017/S0956792500000966
  • [6] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786
  • [7] Franz Peherstorfer and Robert Steinbauer, Characterization of orthogonal polynomials with respect to a functional, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 339–355. MR 1379142, https://doi.org/10.1016/0377-0427(95)00125-5
  • [8] Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725
  • [9] Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088
    Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42C05, 33C05, 39A05

Retrieve articles in all journals with MSC (2010): 42C05, 33C05, 39A05


Additional Information

María José Cantero
Affiliation: Departamento de Matemática Aplicada and IUMA, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, Spain
Email: mjcante@unizar.es

Arieh Iserles
Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom
Email: A.Iserles@damtp.cam.ac.uk

DOI: https://doi.org/10.1090/tran/6454
Received by editor(s): June 27, 2013
Received by editor(s) in revised form: April 3, 2014
Published electronically: September 15, 2015
Additional Notes: The work of the first author was partially supported by the research projects MTM2011-28952-C02-01 and MTM2014-53963-P from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF), and by Project E-64 of Diputación General de Aragón (Spain).
Article copyright: © Copyright 2015 American Mathematical Society