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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.18 Sums

Contents

§18.18(i) Series Expansions of Arbitrary Functions

Jacobi

Let f(z) be analytic within an ellipse E with foci z=±1, and

18.18.1 an=n!(2n+α+β+1)Γ(n+α+β+1)2α+β+1Γ(n+α+1)Γ(n+β+1)-11f(x)Pn(α,β)(x)(1-x)α(1+x)βx.

Then

18.18.2 f(z)=n=0anPn(α,β)(z),

when z lies in the interior of E. Moreover, the series (18.18.2) converges uniformly on any compact domain within E.

Alternatively, assume f(x) is real and continuous and f(x) is piecewise continuous on (-1,1). Assume also the integrals -11(f(x))2(1-x)α(1+x)βx and -11(f(x))2(1-x)α+1(1+x)β+1x converge. Then (18.18.2), with z replaced by x, applies when -1<x<1; moreover, the convergence is uniform on any compact interval within (-1,1).

Chebyshev

See §3.11(ii), or set α=β=±12 in the above results for Jacobi and refer to (18.7.3)–(18.7.6).

Legendre

This is the case α=β=0 of Jacobi. Equation (18.18.1) becomes

18.18.3 an=(n+12)-11f(x)Pn(x)x.

Laguerre

Assume f(x) is real and continuous and f(x) is piecewise continuous on (0,). Assume also 0(f(x))2-xxαx converges. Then

18.18.4 f(x)=n=0bnLn(α)(x),
0<x<,

where

18.18.5 bn=n!Γ(n+α+1)0f(x)Ln(α)(x)-xxαx.

The convergence of the series (18.18.4) is uniform on any compact interval in (0,).

Hermite

Assume f(x) is real and continuous and f(x) is piecewise continuous on (-,). Assume also -(f(x))2-x2x converges. Then

18.18.6 f(x)=n=0dnHn(x),
-<x<,

where

18.18.7 dn=1π2nn!-f(x)Hn(x)-x2x.

The convergence of the series (18.18.6) is uniform on any compact interval in (-,).

§18.18(ii) Addition Theorems

Ultraspherical

18.18.8 Cn(λ)(cosθ1cosθ2+sinθ1sinθ2cosϕ)==0n22(n-)!2λ+2-12λ-1((λ))2(2λ)n+×(sinθ1)Cn-(λ+)(cosθ1)(sinθ2)×Cn-(λ+)(cosθ2)C(λ-12)(cosϕ),
λ>0, λ12.

For the case λ=12 use (18.18.9); compare (18.7.9).

Legendre

18.18.9 Pn(cosθ1cosθ2+sinθ1sinθ2cosϕ)=Pn(cosθ1)Pn(cosθ2)+2=1n(n-)!(n+)!22(n!)2(sinθ1)Pn-(,)(cosθ1)×(sinθ2)Pn-(,)(cosθ2)cos(ϕ).

For (18.18.8), (18.18.9), and the corresponding formula for Jacobi polynomials see Koornwinder (1975b). See also (14.30.9).

Laguerre

18.18.10 Ln(α1++αr+r-1)(x1++xr)=m1++mr=nLm1(α1)(x1)Lmr(αr)(xr).

Hermite

18.18.11 (a12++ar2)12nn!Hn(a1x1++arxr(a12++ar2)12)=m1++mr=na1m1armrm1!mr!Hm1(x1)Hmr(xr).

§18.18(iii) Multiplication Theorems

Laguerre

18.18.12 Ln(α)(λx)Ln(α)(0)==0n(n)λ(1-λ)n-L(α)(x)L(α)(0).

Hermite

18.18.13 Hn(λx)=λn=0n/2(-n)2!(1-λ-2)Hn-2(x).

§18.18(iv) Connection Formulas

Jacobi

18.18.14 Pn(γ,β)(x) =(β+1)n(α+β+2)n=0nα+β+2+1α+β+1(α+β+1)(n+β+γ+1)(β+1)(n+α+β+2)(γ-α)n-(n-)!P(α,β)(x),
18.18.15 (1+x2)n =(β+1)n(α+β+2)n=0nα+β+2+1α+β+1(α+β+1)(n-+1)(β+1)(n+α+β+2)P(α,β)(x),

and a similar pair of equations by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

18.18.16 Cn(μ)(x) ==0n/2λ+n-2λ(μ)n-(λ+1)n-(μ-λ)!Cn-2(λ)(x),
18.18.17 (2x)n =n!=0n/2λ+n-2λ1(λ+1)n-!Cn-2(λ)(x).

Laguerre

Hermite

18.18.20 (2x)n==0n/2(-n)2!Hn-2(x).

§18.18(v) Linearization Formulas

Chebyshev

18.18.21 Tm(x)Tn(x)=12(Tm+n(x)+Tm-n(x)).

Ultraspherical

18.18.22 Cm(λ)(x)Cn(λ)(x)==0min(m,n)(m+n+λ-2)(m+n-2)!(m+n+λ-)!(m-)!(n-)!×(λ)(λ)m-(λ)n-(2λ)m+n-(λ)m+n-(2λ)m+n-2Cm+n-2(λ)(x).

Hermite

18.18.23 Hm(x)Hn(x)==0min(m,n)(m)(n)2!Hm+n-2(x).

The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case λ>0.

§18.18(vi) Bateman-Type Sums

Jacobi

With

18.18.24 bn,=(n)(n+α+β+1)(-β-n)n-2(α+1)n,
18.18.25 Pn(α,β)(x)Pn(α,β)(1)Pn(α,β)(y)Pn(α,β)(1)==0nbn,(x+y)P(α,β)((1+xy)/(x+y))P(α,β)(1),
18.18.26 Pn(α,β)(x)Pn(α,β)(1)==0nbn,(x+1).

§18.18(vii) Poisson Kernels

Laguerre

18.18.27 n=0n!Ln(α)(x)Ln(α)(y)(α+1)nzn=Γ(α+1)(xyz)-12α1-zexp(-(x+y)z1-z)Iα(2(xyz)121-z),
|z|<1.

For the modified Bessel function Iν(z) see §10.25(ii).

Hermite

18.18.28 n=0Hn(x)Hn(y)2nn!zn=(1-z2)-12exp(2xyz-(x2+y2)z21-z2),
|z|<1.

These Poisson kernels are positive, provided that x,y are real, 0z<1, and in the case of (18.18.27) x,y0.

§18.18(viii) Other Sums

In this subsection the variables x and y are not confined to the closures of the intervals of orthogonality; compare §18.2(i).

Ultraspherical

18.18.29 =0nC(λ)(x)Cn-(μ)(x)=Cn(λ+μ)(x).
18.18.30 =0n+2λ2λC(λ)(x)xn-=Cn(λ+1)(x).

Chebyshev

18.18.31 =0nT(x)xn- =Un(x).
18.18.32 2=0nT2(x) =1+U2n(x),
18.18.33 2=0nT2+1(x) =U2n+1(x).
18.18.34 2(1-x2)=0nU2(x) =1-T2n+2(x),
18.18.35 2(1-x2)=0nU2+1(x) =x-T2n+3(x).

Legendre and Chebyshev

Laguerre

18.18.37 =0nL(α)(x)=Ln(α+1)(x),
18.18.38 =0nL(α)(x)Ln-(β)(y)=Ln(α+β+1)(x+y).

Hermite and Laguerre

18.18.39 =0n(n)H(212x)Hn-(212y)=212nHn(x+y),
18.18.40 =0n(n)H2(x)H2n-2(y)=(-1)n22nn!Ln(x2+y2).

§18.18(ix) Compendia

For further sums see Hansen (1975, pp. 292-330), Gradshteyn and Ryzhik (2000, pp. 978–993), and Prudnikov et al. (1986b, pp. 637-644 and 700-718).