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18 Orthogonal PolynomialsAskey Scheme

§18.25 Wilson Class: Definitions


§18.25(i) Preliminaries

For the Wilson class OP’s pn(x) with x=λ(y): if the y-orthogonality set is {0,1,,N}, then the role of the differentiation operator d/dx in the Jacobi, Laguerre, and Hermite cases is played by the operator Δy followed by division by Δy(λ(y)), or by the operator y followed by division by y(λ(y)). Alternatively if the y-orthogonality interval is (0,), then the role of d/dx is played by the operator δy followed by division by δy(λ(y)).

Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials Wn(x;a,b,c,d), continuous dual Hahn polynomials Sn(x;a,b,c), Racah polynomials Rn(x;α,β,γ,δ), and dual Hahn polynomials Rn(x;γ,δ,N).

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
pn(x) x=λ(y) Orthogonality range for y Constraints
Wn(x;a,b,c,d) y2 (0,) (a,b,c,d)>0; nonreal parameters in conjugate pairs
Sn(x;a,b,c) y2 (0,) (a,b,c)>0; nonreal parameters in conjugate pairs
Rn(x;α,β,γ,δ) y(y+γ+δ+1) {0,1,,N} α+1 or β+δ+1 or γ+1=-N; for further constraints see (18.25.1)
Rn(x;γ,δ,N) y(y+γ+δ+1) {0,1,,N} γ,δ>-1 or <-N

Further Constraints for Racah Polynomials

If α+1=-N, then the weights will be positive iff one of the following eight sets of inequalities holds:

18.25.1 -δ-1 <β<γ+1<-N+1.
N-1 <-δ-1<β<γ+1.
γ,δ >-1,β>N+γ.
γ,δ >-1,β<-N-δ.
N-1 <N+γ<β<-N-δ.
N+γ <β<-N-δ<-N-1.
γ,δ <-N,β>-1-δ.
γ,δ <-N,β<γ+1.

The first four sets imply γ+δ>-2, and the last four imply γ+δ<-2N.

§18.25(ii) Weights and Normalizations: Continuous Cases

18.25.2 0pn(x)pm(x)w(x)dx=hnδn,m.


18.25.3 pn(x)=Wn(x;a1,a2,a3,a4),
18.25.4 w(y2)=12y|jΓ(aj+iy)Γ(2iy)|2,
18.25.5 hn=n! 2πj<Γ(n+aj+a)(2n-1+jaj)Γ(n-1+jaj).

Continuous Dual Hahn

18.25.6 pn(x) =Sn(x;a1,a2,a3),
18.25.7 w(y2) =12y|jΓ(aj+iy)Γ(2iy)|2,
18.25.8 hn =n! 2πj<Γ(n+aj+a).

§18.25(iii) Weights and Normalizations: Discrete Cases

18.25.9 y=0Npn(y(y+γ+δ+1))pm(y(y+γ+δ+1))γ+δ+1+2yγ+δ+1+yωy=hnδn,m.


18.25.10 pn(x)=Rn(x;α,β,γ,δ),
18.25.11 ωy =(α+1)y(β+δ+1)y(γ+1)y(γ+δ+2)y(-α+γ+δ+1)y(-β+γ+1)y(δ+1)yy!,
18.25.12 hn =(-β)N(γ+δ+2)N(-β+γ+1)N(δ+1)N(n+α+β+1)nn!(α+β+2)2n(α+β-γ+1)n(α-δ+1)n(β+1)n(α+1)n(β+δ+1)n(γ+1)n.

Dual Hahn

18.25.13 pn(x)=Rn(x;γ,δ,N),
18.25.14 ωy=(-1)y(-N)y(γ+1)y(γ+δ+1)2(N+γ+δ+2)y(δ+1)yy!,
18.25.15 hn=n!(N-n)!(γ+δ+2)NN!(γ+1)n(δ+1)N-n.

§18.25(iv) Leading Coefficients

Table 18.25.2 provides the leading coefficients kn18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials.

Table 18.25.2: Wilson class OP’s: leading coefficients.
pn(x) kn
Wn(x;a,b,c,d) (-1)n(n+a+b+c+d-1)n
Sn(x;a,b,c) (-1)n
Rn(x;α,β,γ,δ) (n+α+β+1)n(α+1)n(β+δ+1)n(γ+1)n
Rn(x;γ,δ,N) 1(γ+1)n(-N)n