What's New
About the Project
NIST
18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.17 Integrals

Contents

§18.17(i) Indefinite Integrals

Jacobi

18.17.1 2n0x(1-y)α(1+y)βPn(α,β)(y)dy=Pn-1(α+1,β+1)(0)-(1-x)α+1(1+x)β+1Pn-1(α+1,β+1)(x).

Laguerre

18.17.2 0xLm(y)Ln(x-y)dy=0xLm+n(y)dy=Lm+n(x)-Lm+n+1(x).

Hermite

18.17.3 0xHn(y)dy=12(n+1)(Hn+1(x)-Hn+1(0)),
18.17.4 0xe-y2Hn(y)dy=Hn-1(0)-e-x2Hn-1(x).

§18.17(ii) Integral Representations for Products

Ultraspherical

18.17.5 Cn(λ)(cosθ1)Cn(λ)(1)Cn(λ)(cosθ2)Cn(λ)(1)=Γ(λ+12)π12Γ(λ)0πCn(λ)(cosθ1cosθ2+sinθ1sinθ2cosϕ)Cn(λ)(1)(sinϕ)2λ-1dϕ,
λ>0.

Legendre

18.17.6 Pn(cosθ1)Pn(cosθ2)=1π0πPn(cosθ1cosθ2+sinθ1sinθ2cosϕ)dϕ.

For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977).

§18.17(iii) Nicholson-Type Integrals

Legendre

18.17.7 (Pn(x))2+4π-2(Qn(x))2=4π-21Qn(x2+(1-x2)t)(t2-1)-12dt,
-1<x<1.

For the Ferrers function Qn(x) and Legendre function Qn(x) see §§14.3(i) and 14.3(ii), with μ=0 and ν=n.

Hermite

18.17.8 (Hn(x))2+2n(n!)2ex2(V(-n-12,212x))2=2n+32n!ex2π0e-(2n+1)t+x2tanht(sinh2t)12dt.

For the parabolic cylinder function V(a,z) see §12.2. For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978).

§18.17(iv) Fractional Integrals

Jacobi

18.17.9 (1-x)α+μPn(α+μ,β-μ)(x)Γ(α+μ+n+1)=x1(1-y)αPn(α,β)(y)Γ(α+n+1)(y-x)μ-1Γ(μ)dy,
μ>0, -1<x<1,
18.17.10 xβ+μ(x+1)nΓ(β+μ+n+1)Pn(α,β+μ)(x-1x+1) =0xyβ(y+1)nΓ(β+n+1)Pn(α,β)(y-1y+1)(x-y)μ-1Γ(μ)dy,
μ>0, x>0,
18.17.11 Γ(n+α+β-μ+1)xn+α+β-μ+1Pn(α,β-μ)(1-2x-1) =xΓ(n+α+β+1)yn+α+β+1Pn(α,β)(1-2y-1)(y-x)μ-1Γ(μ)dy,
α+β+1>μ>0, x>1,

and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

18.17.12 Γ(λ-μ)Cn(λ-μ)(x-12)xλ-μ+12n =xΓ(λ)Cn(λ)(y-12)yλ+12n(y-x)μ-1Γ(μ)dy,
λ>μ>0, x>0,
18.17.13 x12n(x-1)λ+μ-12Γ(λ+μ+12)Cn(λ+μ)(x-12)Cn(λ+μ)(1) =1xy12n(y-1)λ-12Γ(λ+12)Cn(λ)(y-12)Cn(λ)(1)(x-y)μ-1Γ(μ)dy,
μ>0, x>1.

Laguerre

18.17.14 xα+μLn(α+μ)(x)Γ(α+μ+n+1) =0xyαLn(α)(y)Γ(α+n+1)(x-y)μ-1Γ(μ)dy,
μ>0, x>0.
18.17.15 e-xLn(α)(x) =xe-yLn(α+μ)(y)(y-x)μ-1Γ(μ)dy,
μ>0.

§18.17(v) Fourier Transforms

Throughout this subsection we assume y>0; sometimes however, this restriction can be eased by analytic continuation.

Jacobi

18.17.16 -11(1-x)α(1+x)βPn(α,β)(x)eixydx=(iy)neiyn!2n+α+β+1B(n+α+1,n+β+1)F11(n+α+1;2n+α+β+2;-2iy).

For the beta function B(a,b) see §5.12, and for the confluent hypergeometric function F11 see (16.2.1) and Chapter 13.

Ultraspherical

18.17.17 01(1-x2)λ-12C2n(λ)(x)cos(xy)dx=(-1)nπΓ(2n+2λ)Jλ+2n(y)(2n)!Γ(λ)(2y)λ,
18.17.18 01(1-x2)λ-12C2n+1(λ)(x)sin(xy)dx=(-1)nπΓ(2n+2λ+1)J2n+λ+1(y)(2n+1)!Γ(λ)(2y)λ.

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

Legendre

18.17.19 -11Pn(x)eixydx=in2πyJn+12(y),
18.17.20 01Pn(1-2x2)cos(xy)dx=(-1)n12πJn+12(12y)J-n-12(12y),
18.17.21 01Pn(1-2x2)sin(xy)dx=12π(Jn+12(12y))2.

Hermite

18.17.22 12π-e-14x2Hen(x)e12ixydx=ine-14y2Hen(y),
18.17.23 0e-12x2He2n(x)cos(xy)dx=(-1)n12πy2ne-12y2,
18.17.24 0e-x2He2n(2x)cos(xy)dx=(-1)n12πe-14y2He2n(y).
18.17.25 0e-12x2Hen(x)Hen+2m(x)cos(xy)dx=(-1)m12πn!y2me-12y2Ln(2m)(y2),
18.17.26 0e-12x2Hen(x)Hen+2m+1(x)sin(xy)dx=(-1)m12πn!y2m+1e-12y2Ln(2m+1)(y2).
18.17.27 0e-12x2He2n+1(x)sin(xy)dx=(-1)n12πy2n+1e-12y2,
18.17.28 0e-x2He2n+1(2x)sin(xy)dx=(-1)n12πe-14y2He2n+1(y).

Laguerre

18.17.29 0x2me-12x2Ln(2m)(x2)cos(xy)dx=(-1)m12π1n!e-12y2Hen(y)Hen+2m(y).
18.17.30 0x2ne-12x2Ln(n-12)(12x2)cos(xy)dx=12πy2ne-12y2Ln(n-12)(12y2).
18.17.31 0e-axxν-2nL2n-1(ν-2n)(ax)cos(xy)dx=i(-1)nΓ(ν)2(2n-1)!y2n-1((a+iy)-ν-(a-iy)-ν),
ν>2n-1, a>0,
18.17.32 0e-axxν-1-2nL2n(ν-1-2n)(ax)cos(xy)dx=(-1)nΓ(ν)2(2n)!y2n((a+iy)-ν+(a-iy)-ν),
ν>2n, a>0.

§18.17(vi) Laplace Transforms

Jacobi

18.17.33 -11e-(x+1)zPn(α,β)(x)(1-x)α(1+x)βdx=(-1)n2α+β+n+1Γ(α+n+1)Γ(β+n+1)Γ(α+β+2n+2)n!znF11(β+n+1α+β+2n+2;-2z),
z.

For the confluent hypergeometric function F11 see (16.2.1) and Chapter 13.

Laguerre

Hermite

§18.17(vii) Mellin Transforms

Jacobi

18.17.36 -11(1-x)z-1(1+x)βPn(α,β)(x)dx=2β+zΓ(z)Γ(1+β+n)(1+α-z)nn!Γ(1+β+z+n),
z>0.

Ultraspherical

18.17.37 01(1-x2)λ-12Cn(λ)(x)xz-1dx=π 21-2λ-zΓ(n+2λ)Γ(z)n!Γ(λ)Γ(12+12n+λ+12z)Γ(12+12z-12n),
z>0.

Legendre

18.17.38 01P2n(x)xz-1dx=(-1)n(12-12z)n2(12z)n+1,
z>0,
18.17.39 01P2n+1(x)xz-1dx=(-1)n(1-12z)n2(12+12z)n+1,
z>-1.

Laguerre

18.17.40 0e-axLn(α)(bx)xz-1dx=Γ(z+n)n!(a-b)na-n-zF12(-n,1+α-z1-n-z;aa-b),
a>0, z>0.

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

Hermite

18.17.41 0e-axHen(x)xz-1dx=Γ(z+n)a-n-2F22(-12n,-12n+12-12z-12n,-12z-12n+12;-12a2),
a>0. Also, z>0, n even; z>-1, n odd.

For the generalized hypergeometric function F22 see (16.2.1).

§18.17(viii) Other Integrals

Chebyshev

18.17.42 -11Tn(y)(1-y2)-12y-xdy=πUn-1(x),
18.17.43 -11Un-1(y)(1-y2)12y-xdy=-πTn(x).

These integrals are Cauchy principal values (§1.4(v)).

Legendre

18.17.44 -11Pn(x)-Pn(t)|x-t|dt=2(1+12++1n)Pn(x),
-1x1.

The case x=1 is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972).

18.17.45 (n+12)(1+x)12-1x(x-t)-12Pn(t)dt=Tn(x)+Tn+1(x),
18.17.46 (n+12)(1-x)12x1(t-x)-12Pn(t)dt=Tn(x)-Tn+1(x).

Laguerre

18.17.47 0xtαLm(α)(t)Lm(α)(0)(x-t)βLn(β)(x-t)Ln(β)(0)dt=Γ(α+1)Γ(β+1)Γ(α+β+2)xα+β+1Lm+n(α+β+1)(x)Lm+n(α+β+1)(0).

Hermite

18.17.48 -Hm(y)e-y2Hn(x-y)e-(x-y)2dy=π122-12(m+n+1)Hm+n(2-12x)e-12x2.
18.17.49 -H(x)Hm(x)Hn(x)e-x2dx=212(+m+n)!m!n!π(12+12m-12n)!(12m+12n-12)!(12n+12-12m)!,

provided that +m+n is even and the sum of any two of ,m,n is not less than the third; otherwise the integral is zero.

§18.17(ix) Compendia

For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).