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§18.37 Classical OP’s in Two or More Variables

Contents
  1. §18.37(i) Disk Polynomials
  2. §18.37(ii) OP’s on the Triangle
  3. §18.37(iii) OP’s Associated with Root Systems

§18.37(i) Disk Polynomials

Definition in Terms of Jacobi Polynomials

18.37.1 Rm,n(α)(reiθ)=ei(mn)θr|mn|Pmin(m,n)(α,|mn|)(2r21)Pmin(m,n)(α,|mn|)(1),
r0, θ, α>1.

Orthogonality

18.37.2 x2+y2<1Rm,n(α)(x+iy)Rj,(α)(xiy)(1x2y2)αdxdy=0,
mj and/or n.

Equivalent Definition

The following three conditions, taken together, determine Rm,n(α)(z) uniquely:

18.37.3 Rm,n(α)(z)=j=0min(m,n)cjzmjz¯nj,

where cj are real or complex constants, with c00;

18.37.4 x2+y2<1Rm,n(α)(x+iy)(xiy)mj(x+iy)nj(1x2y2)αdxdy=0,
j=1,2,,min(m,n);
18.37.5 Rm,n(α)(1)=1.

Explicit Representation

18.37.6 Rm,n(α)(z)=j=0min(m,n)(1)j(α+1)m+nj(m)j(n)j(α+1)m(α+1)nj!zmjz¯nj.

§18.37(ii) OP’s on the Triangle

Definition in Terms of Jacobi Polynomials

18.37.7 Pm,nα,β,γ(x,y)=Pmn(α,β+γ+2n+1)(2x1)xnPn(β,γ)(2x1y1),
mn0, α,β,γ>1.

Orthogonality

18.37.8 0<y<x<1Pm,nα,β,γ(x,y)Pj,α,β,γ(x,y)(1x)α(xy)βyγdxdy=0,
mj and/or n.

See Dunkl and Xu (2001, §2.3.3) for analogs of (18.37.1) and (18.37.7) on a d-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 q-ultraspherical, or Askey–Wilson polynomials. In several variables they occur, for q=1, 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 q they occur as Macdonald polynomials for root system An, as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992).