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§18.35 Pollaczek Polynomials

Contents

§18.35(i) Definition and Hypergeometric Representation

18.35.1 P-1(λ)(x;a,b) =0,
P0(λ)(x;a,b) =1,

and

18.35.2 (n+1)Pn+1(λ)(x;a,b)=2((n+λ+a)x+b)Pn(λ)(x;a,b)-(n+2λ-1)Pn-1(λ)(x;a,b),
n=0,1,.

Next, let

18.35.3 τa,b(θ)=acosθ+bsinθ,
0<θ<π.

Then

18.35.4 Pn(λ)(cosθ;a,b)=(λ-iτa,b(θ))nn!einθF12(-n,λ+iτa,b(θ)-n-λ+1+iτa,b(θ);e-2iθ)==0n(λ+iτa,b(θ))!(λ-iτa,b(θ))n-(n-)!ei(n-2)θ.

For the hypergeometric function F12 see §§15.1, 15.2(i).

§18.35(ii) Orthogonality

18.35.5 -11Pn(λ)(x;a,b)Pm(λ)(x;a,b)w(λ)(x;a,b)dx=0,
nm,

where

18.35.6 w(λ)(cosθ;a,b)=π-122λ-1e(2θ-π)τa,b(θ)(sinθ)2λ-1|Γ(λ+iτa,b(θ))|2,
ab-a, λ>-12, 0<θ<π.

§18.35(iii) Other Properties

18.35.7 (1-zeiθ)-λ+iτa,b(θ)(1-ze-iθ)-λ-iτa,b(θ)=n=0Pn(λ)(cosθ;a,b)zn,
|z|<1, 0<θ<π.
18.35.8 Pn(λ)(x;0,0)=Cn(λ)(x),
18.35.9 Pn(λ)(cosϕ;0,xsinϕ)=Pn(λ)(x;ϕ).

For the polynomials Cn(λ)(x) and Pn(λ)(x;ϕ) see §§18.3 and 18.19, respectively.

See Bo and Wong (1996) for an asymptotic expansion of Pn(12)(cos(n-12θ);a,b) as n, with a and b fixed. This expansion is in terms of the Airy function Ai(x) and its derivative (§9.2), and is uniform in any compact θ-interval in (0,). Also included is an asymptotic approximation for the zeros of Pn(12)(cos(n-12θ);a,b).