# §18.37 Classical OP’s in Two or More Variables

## §18.37(i) Disk Polynomials

### ¶ Equivalent Definition

The following three conditions, taken together, determine uniquely:

where are real or complex constants, with ;

## §18.37(ii) OP’s on the Triangle

18.37.7, .

### ¶ Orthogonality

See Dunkl and Xu (2001, §2.3.3) for analogs of (18.37.1) and (18.37.7) on a -dimensional simplex.

## §18.37(iii) OP’s Associated with Root Systems

Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. In one variable they are essentially ultraspherical, Jacobi, continuous -ultraspherical, or Askey–Wilson polynomials. In several variables they occur, for , as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). For general they occur as Macdonald polynomials for root system , as Macdonald polynomials for general root systems, and as Macdonald-Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992).