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18 Orthogonal PolynomialsAskey Scheme

§18.26 Wilson Class: Continued

Contents

§18.26(i) Representations as Generalized Hypergeometric Functions

For the definition of generalized hypergeometric functions see §16.2.

18.26.1 Wn(y2;a,b,c,d)=(a+b)n(a+c)n(a+d)nF34(-n,n+a+b+c+d-1,a+iy,a-iya+b,a+c,a+d;1).
18.26.2 Sn(y2;a,b,c)(a+b)n(a+c)n=F23(-n,a+iy,a-iya+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),
n=0,1,,N.

§18.26(ii) Limit Relations

Wilson Continuous Dual Hahn

Wilson Continuous Hahn

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

Wilson Jacobi

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

Continuous Dual Hahn Meixner–Pollaczek

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

Racah Dual Hahn

Racah Hahn

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

Dual Hahn Krawtchouk

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

Dual Hahn Meixner

With

18.26.12 r(x;β,c,N)=x(x+β+c-1(1-c)N),
18.26.13 limNRn(r(x;β,c,N);β-1,c-1(1-c)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+d-1)Wn-1(y2;a+12,b+12,c+12,d+12).
18.26.15 δy(Sn(y2;a,b,c))/δy(y2)=-nSn-1(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)Rn-1(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)NRn-1(y(y+γ+δ+2);γ+1,δ,N-1).

§18.26(iv) Generating Functions

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

Wilson

18.26.18 F12(a+iy,d+iya+d;z)F12(b-iy,c-iyb+c;z)=n=0Wn(y2;a,b,c,d)(a+d)n(b+c)nn!zn,
|z|<1.

Continuous Dual Hahn

Racah

18.26.20 F12(-y,-y+β-γβ+δ+1;z)F12(y-N,y+γ+1-δ-N;z)=n=0N(-N)n(γ+1)n(-δ-N)nn!Rn(y(y+γ+δ+1);-N-1,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.