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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) (1-x)α(1+x)β 𝒜n (n+α+β+1)n2nn! n(α-β)2n+α+β α,β>-1

Ultraspherical

(Gegenbauer)

Cn(λ)(x) (-1,1) (1-x2)λ-12 21-2λπΓ(n+2λ)(n+λ)(Γ(λ))2n! 2n(λ)nn! 0 λ>-12,λ0

Chebyshev

of first kind

Tn(x) (-1,1) (1-x2)-12 {12π,n>0π,n=0 {2n-1,n>01,n=0 0

Chebyshev

of second kind

Un(x) (-1,1) (1-x2)12 12π 2n 0

Chebyshev

of third kind

Vn(x) (-1,1) (1-x)12(1+x)-12 π 2n 12

Chebyshev

of fourth kind

Wn(x) (-1,1) (1-x)-12(1+x)12 π 2n -12

Shifted Chebyshev

of first kind

Tn*(x) (0,1) (x-x2)-12 {12π,n>0π,n=0 {22n-1,n>01,n=0 -12n

Shifted Chebyshev

of second kind

Un*(x) (0,1) (x-x2)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,) e-xxα Γ(n+α+1)/n! (-1)n/n! -n(n+α) α>-1
Hermite Hn(x) (-,) e-x2 π122nn! 2n 0
Hermite Hen(x) (-,) e-12x2 (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 x-1 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).

Chebyshev

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,

where

18.3.2 xN+1,n=cos((n-12)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

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.