# explicit forms

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## 1—10 of 23 matching pages

##### 1: 10.29 Recurrence Relations and Derivatives

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###### §10.29(ii) Derivatives

…##### 2: 10.6 Recurrence Relations and Derivatives

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###### §10.6(ii) Derivatives

…##### 3: 18.40 Methods of Computation

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►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)).
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##### 4: 3.5 Quadrature

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►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. …##### 5: 18.17 Integrals

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►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively.
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##### 6: 18.15 Asymptotic Approximations

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►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.
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##### 7: 19.14 Reduction of General Elliptic Integrals

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►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.
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##### 8: 18.39 Applications in the Physical Sciences

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►Here are three examples of solutions for (18.39.8) for explicit choices of $V(x)$ and with the ${\psi}_{n}(x)$ corresponding to the discrete spectrum.
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►The orthonormal stationary states and corresponding eigenvalues are then of the form
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►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below.
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►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$.
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►These, taken together with the infinite sets of bound states for each $l$, form complete sets.
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##### 9: 29.15 Fourier Series and Chebyshev Series

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►Since (29.2.5) implies that $\mathrm{cos}\varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form
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29.15.50
$${\mathrm{\mathit{s}\mathit{c}\mathit{d}\mathit{E}}}_{2n+3}^{m}(z,{k}^{2})=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^{n}{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

►For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).
##### 10: 18.32 OP’s with Respect to Freud Weights

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►A

*Freud weight*is a weight function of the form …No explicit expressions for the corresponding OP’s are available. … ►*Generalized Freud weights*have the form … ►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty})$, where the term*half-Freud weight*is used; or on $x\in [-1,1]$ or $[0,1]$, where the term*Rys weight*is employed, see Rys et al. (1983). …