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

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- by María José Cantero and Arieh Iserles PDF
- Trans. Amer. Math. Soc.
**368**(2016), 4027-4063 Request permission

## Abstract:

We explore orthogonal polynomials on the unit circle whose Schur parameters are $\{c\alpha ^n\}_{n=1}^\infty$, where $0<|\alpha |,|c|<1$. 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

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## 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
- 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).
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 4027-4063 - MSC (2010): Primary 42C05, 33C05, 39A05
- DOI: https://doi.org/10.1090/tran/6454
- MathSciNet review: 3453364