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18 Orthogonal PolynomialsAskey Scheme

§18.19 Hahn Class: Definitions

Hahn, Krawtchouk, Meixner, and Charlier

Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials Qn(x;α,β,N), Krawtchouk polynomials Kn(x;p,N), Meixner polynomials Mn(x;β,c), and Charlier polynomials Cn(x,a).

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
pn(x) X wx hn

Qn(x;α,β,N),

n=0,1,,N

{0,1,,N}

(α+1)x(β+1)N-xx!(N-x)!,

α,β>-1 or α,β<-N

(-1)n(n+α+β+1)N+1(β+1)nn!(2n+α+β+1)(α+1)n(-N)nN!

If α,β<-N, then (-1)Nwx>0 and (-1)Nhn>0.

Kn(x;p,N),

n=0,1,,N

{0,1,,N}

(Nx)px(1-p)N-x,

0<p<1

(1-pp)n/(Nn)
Mn(x;β,c) {0,1,2,}

(β)xcx/x!,

β>0, 0<c<1

c-nn!(β)n(1-c)β
Cn(x,a) {0,1,2,} ax/x!, a>0 a-nan!
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
pn(x) kn
Qn(x;α,β,N) (n+α+β+1)n(α+1)n(-N)n
Kn(x;p,N) p-n/(-N)n
Mn(x;β,c) (1-c-1)n/(β)n
Cn(x,a) (-a)-n

Continuous Hahn

These polynomials are orthogonal on (-,), and with a>0, b>0 are defined as follows.

18.19.1 pn(x)=pn(x;a,b,a¯,b¯),
18.19.2 w(z;a,b,a¯,b¯)=Γ(a+z)Γ(b+z)Γ(a¯-z)Γ(b¯-z),
18.19.3 w(x)=w(x;a,b,a¯,b¯)=|Γ(a+x)Γ(b+x)|2,
18.19.4 hn=2πΓ(n+a+a¯)Γ(n+b+b¯)|Γ(n+a+b¯)|2(2n+2(a+b)-1)Γ(n+2(a+b)-1)n!,
18.19.5 kn=(n+2(a+b)-1)nn!.

Meixner–Pollaczek

These polynomials are orthogonal on (-,), and are defined as follows.

18.19.6 pn(x)=Pn(λ)(x;ϕ),
18.19.7 w(λ)(z;ϕ)=Γ(λ+z)Γ(λ-z)(2ϕ-π)z,
18.19.8 w(x)=w(λ)(x;ϕ)=|Γ(λ+x)|2(2ϕ-π)x,
λ>0, 0<ϕ<π,
18.19.9 hn =2πΓ(n+2λ)(2sinϕ)2λn!,
kn =(2sinϕ)nn!.