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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.3 Definitions

Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§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, normalizations, 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 exact values of the coefficients of the Jacobi polynomials Pn(α,β)(x), the ultraspherical polynomials Cn(λ)(x), the Chebyshev polynomials Tn(x) and Un(x), the Legendre polynomials Pn(x), the Laguerre polynomials Ln(x), and the Hermite polynomials Hn(x), see Abramowitz and Stegun (1964, pp. 793–801). The Jacobi polynomials are in powers of x1 for n=0,1,,6. The ultraspherical polynomials are in powers of x for n=0,1,,6. The other polynomials are in powers of x for n=0,1,,12. See also §18.5(iv).


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).


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.