# §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 denotes . For further implications of the parameter constraints see the Note in §18.5(iii).
Name Constraints
Jacobi

Ultraspherical

(Gegenbauer)

0

Chebyshev

of first kind

0

Chebyshev

of second kind

0

Chebyshev

of third kind

Chebyshev

of fourth kind

Shifted Chebyshev

of first kind

Shifted Chebyshev

of second kind

Legendre 1 0
Shifted Legendre 1
Laguerre
Hermite 0
Hermite 1 0

For exact values of the coefficients of the Jacobi polynomials , the ultraspherical polynomials , the Chebyshev polynomials and , the Legendre polynomials , the Laguerre polynomials , and the Hermite polynomials , see Abramowitz and Stegun (1964, pp. 793–801). The Jacobi polynomials are in powers of for . The ultraspherical polynomials are in powers of for . The other polynomials are in powers of for . 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 , , are orthogonal on the discrete point set comprising the zeros , of :

where

When the sum in (18.3.1) is . When the sum in (18.3.1) is .

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 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.