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18 Orthogonal PolynomialsAskey Scheme

§18.21 Hahn Class: Interrelations

Contents

§18.21(i) Dualities

Duality of Hahn and Dual Hahn

18.21.1 Qn(x;α,β,N)=Rx(n(n+α+β+1);α,β,N),
n,x=0,1,,N.

For the dual Hahn polynomial Rn(x;γ,δ,N) see §18.25.

Self-Dualities

18.21.2 Kn(x;p,N) =Kx(n;p,N),
n,x=0,1,,N.
Mn(x;β,c) =Mx(n;β,c),
n,x=0,1,2,.
Cn(x,a) =Cx(n,a),
n,x=0,1,2,.

§18.21(ii) Limit Relations and Special Cases

Hahn Krawtchouk

18.21.3 limtQn(x;pt,(1-p)t,N)=Kn(x;p,N).

Hahn Meixner

18.21.4 limNQn(x;b-1,N(c-1-1),N)=Mn(x;b,c).

Hahn Jacobi

18.21.5 limNQn(Nx;α,β,N)=Pn(α,β)(1-2x)Pn(α,β)(1).

Krawtchouk Charlier

18.21.6 limNKn(x;N-1a,N)=Cn(x,a).

Meixner Charlier

18.21.7 limβMn(x;β,a(a+β)-1)=Cn(x,a).

Meixner Laguerre

18.21.8 limc1Mn((1-c)-1x;α+1,c)=Ln(α)(x)Ln(α)(0).

Charlier Hermite

18.21.9 lima(2a)12nCn((2a)12x+a,a)=(-1)nHn(x).

Continuous Hahn Meixner–Pollaczek

18.21.10 limtt-npn(x-t;λ+it,-ttanϕ,λ-it,-ttanϕ)=(-1)n(cosϕ)nPn(λ)(x;ϕ).
18.21.11 pn(x;a,a+12,a,a+12)=2-2n(4a+n)nPn(2a)(2x;12π).

Meixner–Pollaczek Laguerre

18.21.12 limϕ0Pn(12α+12)(-(2ϕ)-1x;ϕ)=Ln(α)(x).

A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1.

See accompanying text
Figure 18.21.1: Askey scheme. The number of free real parameters is zero for Hermite polynomials. It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. (This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify