classical orthogonal polynomials

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1: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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Quadrature

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Integrable Systems

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Riemann–Hilbert Problems

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Radon Transform

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2: 18.39 Physical Applications

§18.39 Physical Applications

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§18.39(i) Quantum Mechanics

… ►The corresponding eigenfunctions are … ► … ►For physical applications of $q$-Laguerre polynomials see §17.17. …

3: 18.40 Methods of Computation

… ►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

4: 18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. …

5: 18.13 Continued Fractions

§18.13 Continued Fractions

… ►See also Cuyt et al. (2008, pp. 91–99).

6: 18.3 Definitions

§18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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►For exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L_n\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). …

7: 18.5 Explicit Representations

§18.5 Explicit Representations

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Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$F(x)$	${\kappa }_n$
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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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8: 18.18 Sums

§18.18 Sums

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§18.18(ii) Addition Theorems

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§18.18(iii) Multiplication Theorems

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§18.18(iv) Connection Formulas

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9: 18.14 Inequalities

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§18.14(i) Upper Bounds

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§18.14(ii) Turán-Type Inequalities

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§18.14(iii) Local Maxima and Minima

… ►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$.

10: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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§18.7(ii) Quadratic Transformations

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§18.7(iii) Limit Relations

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