## Advanced Studies in Pure Mathematics

### Quadratic transformations for orthogonal polynomials in one and two variables

Tom H. Koornwinder

#### 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

Dates
First available in Project Euclid: 21 September 2018

https://projecteuclid.org/ euclid.aspm/1537499432

Digital Object Identifier
doi:10.2969/aspm/07610419

Mathematical Reviews number (MathSciNet)
MR3837928

Zentralblatt MATH identifier
07039309

#### 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