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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.10 Integral Representations

Contents
  1. §18.10(i) Dirichlet–Mehler-Type Integral Representations
  2. §18.10(ii) Laplace-Type Integral Representations
  3. §18.10(iii) Contour Integral Representations
  4. §18.10(iv) Other Integral Representations

§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)(cosϕcosθ)α+12dϕ,
0<θ<π, α>12.

Legendre

18.10.2 Pn(cosθ)=212π0θcos((n+12)ϕ)(cosϕcosθ)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(αβ)Γ(β+12)010π((cos12θ)2r2(sin12θ)2+irsinθcosϕ)n×(1r2)αβ1r2β+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(x2r2+2ixrcosϕ)ner2×r2α+1(sinϕ)2αdϕdr,
α>12.

Hermite

§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)(zc)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) (1x)α(1+x)β z212(zx) (1z)α(1+z)β x ±1 outside C.
Cn(λ)(x) 1 z1 (12xz+z2)λ 0 e±iθ outside C (where x=cosθ).
Tn(x) 1 z1 1xz12xz+z2 0
Un(x) 1 z1 (12xz+z2)1 0
Pn(x) 1 z1 (12xz+z2)12 0
Pn(x) 1 z212(zx) 1 x
Ln(α)(x) exxα z(zx)1 zαez x 0 outside C.
Hn(x)/n! 1 z1 e2xzz2 0
Hen(x)/n! 1 z1 exz12z2 0

§18.10(iv) Other Integral Representations

Laguerre

18.10.9 Ln(α)(x)=exx12αn!0ettn+12αJα(2xt)dt,
α>1.

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

Hermite

18.10.10 Hn(x)=(2i)nex2π12et2tne2ixtdt=2n+1π12ex20et2tncos(2xt12nπ)dt.

See also §18.17.