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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.12 Generating Functions

With the notation of §§10.2(ii), 10.25(ii), and 15.2,

Jacobi

18.12.1 2α+βR(1+R-z)α(1+R+z)β=n=0Pn(α,β)(x)zn,
R=1-2xz+z2, |z|<1.
18.12.2 (12(1-x)z)-12αJα(2(1-x)z)(12(1+x)z)-12βIβ(2(1+x)z)=n=0Pn(α,β)(x)Γ(n+α+1)Γ(n+β+1)zn.
18.12.3 (1+z)-α-β-1F12(12(α+β+1),12(α+β+2)β+1;2(x+1)z(1+z)2)=n=0(α+β+1)n(β+1)nPn(α,β)(x)zn,
|z|<1,

and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function F12 see §§15.1, 15.2(i).

Ultraspherical

18.12.4 (1-2xz+z2)-λ=n=0Cn(λ)(x)zn=n=0(2λ)n(λ+12)nPn(λ-12,λ-12)(x)zn,
|z|<1.
18.12.5 1-xz(1-2xz+z2)λ+1=n=0n+2λ2λCn(λ)(x)zn,
|z|<1.
18.12.6 Γ(λ+12)ezcosθ(12zsinθ)12-λJλ-12(zsinθ)=n=0Cn(λ)(cosθ)(2λ)nzn,
0θπ.

Chebyshev

18.12.7 1-z21-2xz+z2 =1+2n=1Tn(x)zn,
|z|<1.
18.12.8 1-xz1-2xz+z2 =n=0Tn(x)zn,
|z|<1.
18.12.9 -ln(1-2xz+z2)=2n=1Tn(x)nzn,
|z|<1.
18.12.10 11-2xz+z2=n=0Un(x)zn,
|z|<1.

Legendre

18.12.11 11-2xz+z2=n=0Pn(x)zn,
|z|<1.

Laguerre

18.12.13 (1-z)-α-1exp(xzz-1)=n=0Ln(α)(x)zn,
|z|<1.
18.12.14 Γ(α+1)(xz)-12αezJα(2xz)=n=0Ln(α)(x)(α+1)nzn.

Hermite

18.12.15 e2xz-z2=n=0Hn(x)n!zn,
18.12.16 exz-12z2=n=0Hen(x)n!zn.