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1: 18.40 Methods of Computation
Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). …
2: 10.29 Recurrence Relations and Derivatives
§10.29(ii) Derivatives
3: 10.6 Recurrence Relations and Derivatives
§10.6(ii) Derivatives
4: 18.17 Integrals
5: 3.5 Quadrature
The p n ( x ) are the monic Legendre polynomials, that is, the polynomials P n ( x ) 18.3) scaled so that the coefficient of the highest power of x in their explicit forms is unity. …
6: 18.15 Asymptotic Approximations
The asymptotic behavior of the classical OP’s as x ± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …
7: 19.14 Reduction of General Elliptic Integrals
The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …
8: 18.32 OP’s with Respect to Freud Weights
A Freud weight is a weight function of the form …No explicit expressions for the corresponding OP’s are available. …
9: 29.15 Fourier Series and Chebyshev Series
Since (29.2.5) implies that cos ϕ = sn ( z , k ) , (29.15.1) can be rewritten in the form For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).
10: 3.8 Nonlinear Equations
Sometimes the equation takes the form
§3.8(iv) Zeros of Polynomials
Explicit formulas for the zeros are available if n 4 ; see §§1.11(iii) and 4.43. No explicit general formulas exist when n 5 . …