Advanced Studies in Pure Mathematics

Quadratic transformations for orthogonal polynomials in one and two variables

Tom H. Koornwinder

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Abstract

We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system $BC_2$, i.e., $BC_2$-type Jacobi polynomials if $q=1$ and Koornwinder polynomials in two variables in the $q$-case.

Article information

Source
Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 419-447

Dates
Received: 31 December 2015
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499432

Digital Object Identifier
doi:10.2969/aspm/07610419

Mathematical Reviews number (MathSciNet)
MR3837928

Zentralblatt MATH identifier
07039309

Subjects
Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 33C52: Orthogonal polynomials and functions associated with root systems 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

Keywords
quadratic transformations orthogonal polynomials ($q$-)Askey scheme orthogonal polynomials in two variables $BC_2$-type Jacobi polynomials Koornwinder polynomials in two variables

Citation

Koornwinder, Tom H. Quadratic transformations for orthogonal polynomials in one and two variables. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 419--447, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610419. https://projecteuclid.org/euclid.aspm/1537499432


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