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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.12 Generating Functions

The z-radii of convergence will depend on x, and in first instance we will assume x[1,1] for Jacobi, ultraspherical, Chebyshev and Legendre, x[0,) for Laguerre, and x for Hermite. With the notation of §§10.2(ii), 10.25(ii), 15.2, and 16.2,


18.12.1 2α+βR(1+Rz)α(1+R+z)β=n=0Pn(α,β)(x)zn,
R=12xz+z2, |z|<1,
18.12.2 𝐅10(α+1;(x1)z2)𝐅10(β+1;(x+1)z2)=(12(1x)z)12αJα(2(1x)z)(12(1+x)z)12βIβ(2(1+x)z)=n=0Pn(α,β)(x)Γ(n+α+1)Γ(n+β+1)zn,
18.12.2_5 F12(γ,α+β+1γα+1;1Rz2)F12(γ,α+β+1γβ+1;1R+z2)=n=0(γ)n(α+β+1γ)n(α+1)n(β+1)nPn(α,β)(x)zn,
R=12xz+z2, |z|<1,

with γ arbitrary. Note that (18.12.2_5) yields (18.12.1) by putting γ=0 and (18.12.2) by replacing z by γ2z and next letting γ.

18.12.3 (1+z)αβ1F12(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,
18.12.3_5 1+z(12xz+z2)β+32=n=0(2β+2)n(β+1)nPn(β+1,β)(x)zn,

and similar formulas as (18.12.3) and (18.12.3_5) by symmetry; compare the second row in Table 18.6.1. See Ismail (2009, (4.3.2)) for another variant of (18.12.3).


18.12.4 (12xz+z2)λ=n=0Cn(λ)(x)zn=n=0(2λ)n(λ+12)nPn(λ12,λ12)(x)zn,
18.12.5 1xz(12xz+z2)λ+1=n=0n+2λ2λCn(λ)(x)zn,


18.12.7 1z212xz+z2 =1+2n=1Tn(x)zn,
18.12.8 1xz12xz+z2 =n=0Tn(x)zn,
18.12.9 ln(12xz+z2)=2n=1Tn(x)nzn,
18.12.10 112xz+z2=n=0Un(x)zn,


18.12.11 112xz+z2=n=0Pn(x)zn,
18.12.12 exzJ0(z1x2)=n=0Pn(x)n!zn.


18.12.13 (1z)α1exp(xzz1)=n=0Ln(α)(x)zn,
18.12.14 Γ(α+1)(xz)12αezJα(2xz)=n=0Ln(α)(x)(α+1)nzn.


18.12.15 e2xzz2=n=0Hn(x)n!zn,
18.12.16 exz12z2=n=0𝐻𝑒n(x)n!zn,
18.12.17 1+2xz+4z2(1+4z2)32exp(4x2z21+4z2)=n=0Hn(x)n/2!zn,

See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.