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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.10 Integral Representations

Contents

§18.10(i) Dirichlet-Mehler-Type Integral Representations

Ultraspherical

18.10.1 Pn(α,α)(cosθ)Pn(α,α)(1)=Cn(α+12)(cosθ)Cn(α+12)(1)=2α+12Γ(α+1)π12Γ(α+12)(sinθ)-2α0θcos((n+α+12)ϕ)(cosϕ-cosθ)-α+12dϕ,
0<θ<π, α>-12.

Legendre

18.10.2 Pn(cosθ)=212π0θcos((n+12)ϕ)(cosϕ-cosθ)12dϕ,
0<θ<π.

Generalizations of (18.10.1) are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)).

§18.10(ii) Laplace-Type Integral Representations

Jacobi

18.10.3 Pn(α,β)(cosθ)Pn(α,β)(1)=2Γ(α+1)π12Γ(α-β)Γ(β+12)010π((cos12θ)2-r2(sin12θ)2+irsinθcosϕ)n×(1-r2)α-β-1r2β+1(sinϕ)2βdϕdr,
α>β>-12.

Ultraspherical

18.10.4 Pn(α,α)(cosθ)Pn(α,α)(1)=Cn(α+12)(cosθ)Cn(α+12)(1)=Γ(α+1)π12Γ(α+12)0π(cosθ+isinθcosϕ)n(sinϕ)2αdϕ,
α>-12.

Legendre

18.10.5 Pn(cosθ)=1π0π(cosθ+isinθcosϕ)ndϕ.

Laguerre

18.10.6 Ln(α)(x2)=2(-1)nπ12Γ(α+12)n!00π(x2-r2+2ixrcosϕ)ne-r2×r2α+1(sinϕ)2αdϕdr,
α>-12.

Hermite

18.10.7 Hn(x)=2nπ12-(x+it)ne-t2dt.

§18.10(iii) Contour Integral Representations

Table 18.10.1 gives contour integral representations of the form

18.10.8 pn(x)=g0(x)2πiC(g1(z,x))ng2(z,x)(z-c)-1dz

for the Jacobi, Laguerre, and Hermite polynomials. Here C is a simple closed contour encircling z=c once in the positive sense.

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
pn(x) g0(x) g1(z,x) g2(z,x) c Conditions
Pn(α,β)(x) (1-x)-α(1+x)-β z2-12(z-x) (1-z)α(1+z)β x ±1 outside C.
Cn(λ)(x) 1 z-1 (1-2xz+z2)-λ 0 e±iθ outside C (where x=cosθ).
Tn(x) 1 z-1 1-xz1-2xz+z2 0
Un(x) 1 z-1 (1-2xz+z2)-1 0
Pn(x) 1 z-1 (1-2xz+z2)-12 0
Pn(x) 1 z2-12(z-x) 1 x
Ln(α)(x) exx-α z(z-x)-1 zαe-z x 0 outside C.
Hn(x)/n! 1 z-1 e2xz-z2 0
Hen(x)/n! 1 z-1 exz-12z2 0

§18.10(iv) Other Integral Representations

Laguerre

18.10.9 Ln(α)(x)=exx-12αn!0e-ttn+12αJα(2xt)dt,
α>-1.

For the Bessel function Jν(z) see §10.2(ii).

Hermite

18.10.10 Hn(x)=(-2i)nex2π12-e-t2tne2ixtdt=2n+1π12ex20e-t2tncos(2xt-12nπ)dt.

See also §18.17.