From orthogonal polynomials on the unit circle to functional equations via generating functions
Year: 2015
Type: article
Abstract: We explore orthogonal polynomials on the unit circle whose Schur parameters are { c α n } n = 1 ∞ \{c\alpha ^n\}_{n=1}^\infty , where 0 > | α | , | c | > 1 0>|\alpha |,|c|>1 . Specifically, we derive two different generating functions. The first can be represented explicitly in terms of sums of a q 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.
DOI: 10.1090/tran/6454
Authors: María Cantero, Arieh Iserles
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1785774343