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18 Orthogonal PolynomialsAskey Scheme

§18.20 Hahn Class: Explicit Representations


§18.20(i) Rodrigues Formulas

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).

Hahn, Krawtchouk, Meixner, and Charlier

18.20.1 pn(x)=1κnwxxn(wx=0n-1F(x+)),

In (18.20.1) X and wx are as in Table 18.19.1. For the Hahn polynomials pn(x)=Qn(x;α,β,N) and

18.20.2 F(x) =(x+α+1)(x-N),
κn =(-N)n(α+1)n.

For the Krawtchouk, Meixner, and Charlier polynomials, F(x) and κn are as in Table 18.20.1.

Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).
pn(x) F(x) κn
Kn(x;p,N) x-N (-N)n
Mn(x;β,c) x+β (β)n
Cn(x;a) 1 1

Continuous Hahn

18.20.3 w(x;a,b,a¯,b¯)pn(x;a,b,a¯,b¯)=1n!δxn(w(x;a+12n,b+12n,a¯+12n,b¯+12n)).


18.20.4 w(λ)(x;ϕ)Pn(λ)(x;ϕ)=1n!δxn(w(λ+12n)(x;ϕ)).

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

For the definition of hypergeometric and generalized hypergeometric functions see §16.2.

18.20.5 Qn(x;α,β,N)=F23(-n,n+α+β+1,-xα+1,-N;1),
18.20.6 Kn(x;p,N) =F12(-n,-x-N;p-1),
18.20.7 Mn(x;β,c) =F12(-n,-xβ;1-c-1).
18.20.8 Cn(x;a) =F02(-n,-x-;-a-1).
18.20.9 pn(x;a,b,a¯,b¯)=in(a+a¯)n(a+b¯)nn!F23(-n,n+2(a+b)-1,a+ixa+a¯,a+b¯;1).

(For symmetry properties of pn(x;a,b,a¯,b¯) with respect to a, b, a¯, b¯ see Andrews et al. (1999, Corollary 3.3.4).)

18.20.10 Pn(λ)(x;ϕ)=(2λ)nn!einϕF12(-n,λ+ix2λ;1-e-2iϕ).