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18 Orthogonal PolynomialsOther Orthogonal Polynomials

§18.37 Classical OP’s in Two or More Variables


§18.37(i) Disk Polynomials

Definition in Terms of Jacobi Polynomials

18.37.1 Rm,n(α)(reiθ)=ei(m-n)θr|m-n|Pmin(m,n)(α,|m-n|)(2r2-1)Pmin(m,n)(α,|m-n|)(1),
r0, θ, α>-1.


18.37.2 x2+y2<1Rm,n(α)(x+iy)Rj,(α)(x-iy)(1-x2-y2)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)cjzm-jz¯n-j,

where cj are real or complex constants, with c00;

18.37.4 x2+y2<1Rm,n(α)(x+iy)(x-iy)m-j(x+iy)n-j(1-x2-y2)αdxdy=0,
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+n-j(-m)j(-n)j(α+1)m(α+1)nj!zm-jz¯n-j.

§18.37(ii) OP’s on the Triangle

Definition in Terms of Jacobi Polynomials

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


18.37.8 0<y<x<1Pm,nα,β,γ(x,y)Pj,α,β,γ(x,y)(1-x)α(x-y)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).