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classical orthogonal polynomials

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1: 18.38 Mathematical Applications
§18.38(i) Classical OP’s: Numerical Analysis
Quadrature
Integrable Systems
Riemann–Hilbert Problems
Radon Transform
2: 18.39 Physical Applications
§18.39 Physical Applications
§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 ( x ) ( = P n ( x ) ) see §14.33. …
5: 18.3 Definitions
§18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). …
6: 18.13 Continued Fractions
§18.13 Continued Fractions
See also Cuyt et al. (2008, pp. 91–99).
7: 18.5 Explicit Representations
§18.5 Explicit Representations
Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
p n ( x ) F ( x ) κ n
§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
8: 18.14 Inequalities
§18.14(i) Upper Bounds
§18.14(ii) Turán-Type Inequalities
§18.14(iii) Local Maxima and Minima
The successive maxima of | H n ( x ) | form a decreasing sequence for x 0 , and an increasing sequence for x 0 .
9: 18.7 Interrelations and Limit Relations
§18.7 Interrelations and Limit Relations
§18.7(i) Linear Transformations
§18.7(ii) Quadratic Transformations
§18.7(iii) Limit Relations
10: 18.16 Zeros
Inequalities
Asymptotic Behavior
when α ( - 1 2 , 1 2 ) . … Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of L n ( ± 1 2 ) ( x ) lead immediately to results for the zeros of H n ( x ) . … For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a).