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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.18 Sums

  1. §18.18(i) Series Expansions of Arbitrary Functions
  2. §18.18(ii) Addition Theorems
  3. §18.18(iii) Multiplication Theorems
  4. §18.18(iv) Connection and Inversion Formulas
  5. §18.18(v) Linearization Formulas
  6. §18.18(vi) Bateman-Type Sums
  7. §18.18(vii) Poisson Kernels
  8. §18.18(viii) Other Sums
  9. §18.18(ix) Compendia

§18.18(i) Series Expansions of Arbitrary Functions


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)(1x)α(1+x)βdx.


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(1x)α(1+x)βdx and 11(f(x))2(1x)α+1(1+x)β+1dx 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).


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


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

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


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

18.18.4 f(x)=n=0bnLn(α)(x),


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

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


Assume f(x) is real and continuous and f(x) is piecewise continuous on (,). Assume also (f(x))2ex2dx converges. Then

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


18.18.7 dn=1π2nn!f(x)Hn(x)ex2dx.

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

Expansion of L2 functions

In all three cases of Jacobi, Laguerre and Hermite, if f(x) is L2 on the corresponding interval with respect to the corresponding weight function and if an,bn,dn are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in L2 sense. See Szegő (1975, Theorems 3.1.5 and 5.7.1). See also (18.2.24), (18.2.25).

§18.18(ii) Addition Theorems


18.18.8 Cn(λ)(cosθ1cosθ2+sinθ1sinθ2cosϕ)==0n22(n)!2λ+212λ1((λ))2(2λ)n+×(sinθ1)Cn(λ+)(cosθ1)(sinθ2)×Cn(λ+)(cosθ2)C(λ12)(cosϕ),
λ>0, λ12.

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


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 integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. For (18.18.8) see also (14.30.9). For formulas for Jacobi and Laguerre polynomials analogous to (18.18.8) and (18.18.9), see (Koornwinder, 1975b, 1977).


18.18.10 Ln(α1++αr+r1)(x1++xr)=m1++mr=nLm1(α1)(x1)Lmr(αr)(xr).


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


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


18.18.13 Hn(λx)=λn=0n/2(n)2!(1λ2)Hn2(x).

§18.18(iv) Connection and Inversion Formulas


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)(γα)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. See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of Pn(γ,δ)(x) in terms of Pn(α,β)(x).


18.18.16 Cn(μ)(x) ==0n/2λ+n2λ(μ)n(λ+1)n(μλ)!Cn2(λ)(x),
18.18.17 (2x)n =n!=0n/2λ+n2λ1(λ+1)n!Cn2(λ)(x).

See (18.5.11) for the limit case λ0 of (18.18.16).


18.18.18 Ln(β)(x) ==0n(βα)n(n)!L(α)(x),
18.18.19 xn =(α+1)n=0n(n)(α+1)L(α)(x).


18.18.20 (2x)n==0n/2(n)2!Hn2(x).

§18.18(v) Linearization Formulas


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


18.18.22 Cm(λ)(x)Cn(λ)(x)==0min(m,n)(m+n+λ2)(m+n2)!(m+n+λ)!(m)!(n)!×(λ)(λ)m(λ)n(2λ)m+n(λ)m+n(2λ)m+n2Cm+n2(λ)(x).


18.18.23 Hm(x)Hn(x)==0min(m,n)(m)(n)2!Hm+n2(x).

The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case λ>0. See (18.17.45) and (18.17.49) for integrated forms of (18.18.22) and (18.18.23), respectively. See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials.

§18.18(vi) Bateman-Type Sums



18.18.24 bn,=(n)(n+α+β+1)(βn)n2(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

See (18.2.41) for the Poisson kernel in case of general OP’s.


18.18.27 n=0n!Ln(α)(x)Ln(α)(y)(α+1)nzn=Γ(α+1)(xyz)12α1zexp((x+y)z1z)Iα(2(xyz)121z),

For the modified Bessel function Iν(z) see §10.25(ii). Formula (18.18.27) is known as the Hille–Hardy formula.


18.18.28 n=0Hn(x)Hn(y)2nn!zn=(1z2)12exp(2xyz(x2+y2)z21z2),

Formula (18.18.28) is known as the Mehler formula. See Ismail (2009, Theorem 4.7.2) for a formula called Kibble–Slepian formula, which generalizes (18.18.28).

These Poisson kernels are positive, provided that x,y are real, 0z<1, and in the case of (18.18.27) x,y0. For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938).

§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).


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).


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(1x2)=0nU2(x) =1T2n+2(x),
18.18.35 2(1x2)=0nU2+1(x) =xT2n+3(x).

Legendre and Chebyshev

18.18.36 =0nP(x)Pn(x)=Un(x).


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)H2n2(y)=(1)n22nn!Ln(x2+y2).

See also (18.38.3) for a finite sum of Jacobi polynomials.

§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).