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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.3 Definitions

The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. There are many ways of characterizing the classical OP’s within the general OP’s {pn(x)}, see Al-Salam (1990). The three most important characterizations are:


As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations A(x)pn′′(x)+B(x)pn(x)+λnpn(x)=0, in Table 18.8.1.


With the property that {pn+1(x)}n=0 is again a system of OP’s. See §18.9(iii).


As given by a Rodrigues formula (18.5.5).

Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre.

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. In the second row 𝒜n denotes 2α+β+1Γ(n+α+1)Γ(n+β+1)/((2n+α+β+1)Γ(n+α+β+1)n!), with 𝒜0=2α+β+1Γ(α+1)Γ(β+1)/Γ(α+β+2). For further implications of the parameter constraints see the Note in §18.5(iii).
Name pn(x) (a,b) w(x) hn kn k~n/kn Constraints
Jacobi Pn(α,β)(x) (1,1) (1x)α(1+x)β 𝒜n (n+α+β+1)n2nn! n(αβ)2n+α+β α,β>1
Ultraspherical (Gegenbauer) Cn(λ)(x) (1,1) (1x2)λ12 212λπΓ(n+2λ)(n+λ)(Γ(λ))2n! 2n(λ)nn! 0 λ>12,λ0
Chebyshev of first kind Tn(x) (1,1) (1x2)12 {12π,n>0π,n=0 {2n1,n>01,n=0 0
Chebyshev of second kind Un(x) (1,1) (1x2)12 12π 2n 0
Chebyshev of third kind Vn(x) (1,1) (1x)12(1+x)12 π 2n 12
Chebyshev of fourth kind Wn(x) (1,1) (1x)12(1+x)12 π 2n 12
Shifted Chebyshev of first kind Tn(x) (0,1) (xx2)12 {12π,n>0π,n=0 {22n1,n>01,n=0 12n
Shifted Chebyshev of second kind Un(x) (0,1) (xx2)12 18π 22n 12n
Legendre Pn(x) (1,1) 1 2/(2n+1) 2n(12)n/n! 0
Shifted Legendre Pn(x) (0,1) 1 1/(2n+1) 22n(12)n/n! 12n
Laguerre Ln(α)(x) (0,) exxα Γ(n+α+1)/n! (1)n/n! n(n+α) α>1
Hermite Hn(x) (,) ex2 π122nn! 2n 0
Hermite 𝐻𝑒n(x) (,) e12x2 (2π)12n! 1 0

For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of x1 for Jacobi polynomials, in powers of x for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for n=0,1,,6 are given in §18.5(iv). For explicit power series coefficients up to n=12 for these polynomials and for coefficients up to n=6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801).


In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6).

In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials Tn(x), n=0,1,,N, are orthogonal on the discrete point set comprising the zeros xN+1,n,n=1,2,,N+1, of TN+1(x):

18.3.1 n=1N+1Tj(xN+1,n)Tk(xN+1,n)=0,
0jN, 0kN, jk,


18.3.2 xN+1,n=cos((n12)π/(N+1)).

When j=k0 the sum in (18.3.1) is 12(N+1). When j=k=0 the sum in (18.3.1) is N+1.

For proofs of these results and for similar properties of the Chebyshev polynomials of the second, third, and fourth kinds see Mason and Handscomb (2003, §4.6).

For another version of the discrete orthogonality property of the polynomials Tn(x) see (3.11.9).

Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007).


Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). In consequence, additional properties are included in Chapter 14.

Jacobi on Other Intervals

For 1β>α>1 a finite system of Jacobi polynomials Pn(α,β)(x) is orthogonal on (1,) with weight function w(x)=(x1)α(x+1)β. For ν and N>12 a finite system of Jacobi polynomials Pn(N1+iν,N1iν)(ix) (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on (,) with w(x)=(1+x2)N1e2νarctanx. For further details see Koekoek et al. (2010, (9.8.3) and §9.9).

Bessel polynomials

Bessel polynomials are often included among the classical OP’s. However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. See §18.34.