## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

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

#### 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**

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