explicit forms

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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

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§10.29(ii) Derivatives

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3: 10.6 Recurrence Relations and Derivatives

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§10.6(ii) Derivatives

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4: 18.17 Integrals

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5: 3.5 Quadrature

… ►The $p_n(x)$ are the monic Legendre polynomials, that is, the polynomials $P_n\left(x\right)$ (§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 $x\to \pm \mathrm{\infty }$ 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 $\cos \varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form … ►

29.15.50 $${\mathit{scdE}}_{2n+3}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathit{scdE}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E50

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See also:

Annotations for §29.15(ii), §29.15 and Ch.29

►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 $n\le 4$; see §§1.11(iii) and 4.43. No explicit general formulas exist when $n\ge 5$. …