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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

18.23.1 F11(-xα+1;-z)F11(x-Nβ+1;z)=n=0N(-N)n(β+1)nn!Qn(x;α,β,N)zn,
x=0,1,,N.
18.23.2 F02(-x,-x+β+N+1-;-z)F02(x-N,x+α+1-;z)=n=0N(-N)n(α+1)nn!Qn(x;α,β,N)zn,
x=0,1,,N.

Krawtchouk

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

Meixner

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

Charlier

18.23.5 ez(1-za)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 (1-eiϕz)-λ+ix(1-e-iϕz)-λ-ix=n=0Pn(λ)(x;ϕ)zn,
|z|<1.