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18 Orthogonal PolynomialsAskey Scheme

§18.26 Wilson Class: Continued

  1. §18.26(i) Representations as Generalized Hypergeometric Functions and Dualities
  2. §18.26(ii) Limit Relations
  3. §18.26(iii) Difference Relations
  4. §18.26(iv) Generating Functions
  5. §18.26(v) Asymptotic Approximations

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

For the definition of generalized hypergeometric functions see §16.2. Here we use as convention for (16.2.1) with bq=N, a1=n, and n=0,1,,N that the summation on the right-hand side ends at k=n.

18.26.1 Wn(y2;a,b,c,d)=(a+b)n(a+c)n(a+d)nF34(n,n+a+b+c+d1,a+iy,aiya+b,a+c,a+d;1).
18.26.2 Sn(y2;a,b,c)=(a+b)n(a+c)nF23(n,a+iy,aiya+b,a+c;1).
18.26.3 Rn(y(y+γ+δ+1);α,β,γ,δ)=F34(n,n+α+β+1,y,y+γ+δ+1α+1,β+δ+1,γ+1;1),
α+1 or β+δ+1 or γ+1=N; n=0,1,,N.
18.26.4 Rn(y(y+γ+δ+1);γ,δ,N)=F23(n,y,y+γ+δ+1γ+1,N;1),


18.26.4_1 Rn(y(y+γ+δ+1);γ,δ,N)=Qy(n;γ,δ,N),

compare (18.21.1).

18.26.4_2 Rn(y(y+γ+δ+1);α,β,γ,δ)=Ry(n(n+α+β+1);γ,δ,α,β).

§18.26(ii) Limit Relations

Wilson Continuous Dual Hahn

Wilson Continuous Hahn

18.26.6 limtWn((x+t)2;ait,bit,a¯+it,b¯+it)(2t)nn!=pn(x;a,b,a¯,b¯).

Wilson Jacobi

18.26.7 limtWn(12(1x)t2;12α+12,12α+12,12β+12+it,12β+12it)t2nn!=Pn(α,β)(x).

Continuous Dual Hahn Meixner–Pollaczek

18.26.8 limtSn((xt)2;λ+it,λit,tcotϕ)/tn=n!(cscϕ)nPn(λ)(x;ϕ).

Racah Dual Hahn

18.26.9 limβRn(x;N1,β,γ,δ)=Rn(x;γ,δ,N).

Racah Hahn

18.26.10 limδRn(x(x+γ+δ+1);α,β,N1,δ)=Qn(x;α,β,N).

Dual Hahn Krawtchouk

18.26.11 limtRn(x(x+t+1);pt,(1p)t,N)=Kn(x;p,N).

Dual Hahn Meixner


18.26.12 r(x;β,c,N)=x(x+β+c1(1c)N),
18.26.13 limNRn(r(x;β,c,N);β1,c1(1c)N,N)=Mn(x;β,c).

See also Figure 18.21.1.

§18.26(iii) Difference Relations

For comments on the use of the forward-difference operator Δx, the backward-difference operator x, and the central-difference operator δx, see §18.2(ii).

For each family only the y-difference that lowers n is given. See Koekoek et al. (2010, Chapter 9) for further formulas.

18.26.14 δy(Wn(y2;a,b,c,d))/δy(y2)=n(n+a+b+c+d1)Wn1(y2;a+12,b+12,c+12,d+12).
18.26.15 δy(Sn(y2;a,b,c))/δy(y2)=nSn1(y2;a+12,b+12,c+12).
18.26.16 Δy(Rn(y(y+γ+δ+1);α,β,γ,δ))Δy(y(y+γ+δ+1))=n(n+α+β+1)(α+1)(β+δ+1)(γ+1)Rn1(y(y+γ+δ+2);α+1,β+1,γ+1,δ).
18.26.17 Δy(Rn(y(y+γ+δ+1);γ,δ,N))Δy(y(y+γ+δ+1))=n(γ+1)NRn1(y(y+γ+δ+2);γ+1,δ,N1).

§18.26(iv) Generating Functions

For the hypergeometric function F12 see §§15.1 and 15.2(i).


18.26.18 F12(a+iy,d+iya+d;z)F12(biy,ciyb+c;z)=n=0Wn(y2;a,b,c,d)(a+d)n(b+c)nn!zn,

Continuous Dual Hahn


18.26.20 F12(y,y+βγβ+δ+1;z)F12(yN,y+γ+1δN;z)=n=0N(N)n(γ+1)n(δN)nn!Rn(y(y+γ+δ+1);N1,β,γ,δ)zn.

Dual Hahn

§18.26(v) Asymptotic Approximations

For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998).

Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.