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18 Orthogonal PolynomialsAskey Scheme

§18.23 Hahn Class: Generating Functions

For the definition of generalized hypergeometric functions see §16.2.

Hahn

Krawtchouk

18.23.3 (11ppz)x(1+z)Nx=n=0N(Nn)Kn(x;p,N)zn,
x=0,1,,N.

Meixner

18.23.4 (1zc)x(1z)xβ=n=0(β)nn!Mn(x;β,c)zn,
x=0,1,2,, |z|<1.

Charlier

18.23.5 ez(1za)x=n=0Cn(x;a)n!zn,
x=0,1,2,.

Continuous Hahn

18.23.6 F11(a+ix2a;iz)F11(b¯ix2b;iz)=n=0pn(x;a,b,a¯,b¯)(2a)n(2b)nzn.

Meixner–Pollaczek

18.23.7 (1eiϕz)λ+ix(1eiϕz)λix=n=0Pn(λ)(x;ϕ)zn,
|z|<1.