# §18.5 Explicit Representations

## §18.5(ii) Rodrigues Formulas

18.5.5

In this equation is as in Table 18.3.1, and , are as in Table 18.5.1.

## §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

For the definitions of , , and see §16.2.

### ¶ Hermite

For corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11).

Note. The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of when the conditions and are not satisfied. However, in these circumstances the orthogonality property (18.2.1) disappears. For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials we assume throughout this chapter that and , unless stated otherwise. Similarly in the cases of the ultraspherical polynomials and the Laguerre polynomials we assume that , and , unless stated otherwise.

## §18.5(iv) Numerical Coefficients

### ¶ Hermite

18.5.18

For the corresponding polynomials of degrees 7 through 12 see Abramowitz and Stegun (1964, Tables 22.3, 22.5, 22.9, 22.10, 22.12).