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1: 18.40 Methods of Computation
Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OPs of large degree. However, for applications in which the OPs 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 OPs other than Chebyshev. …
2: 18.32 OP’s with Respect to Freud Weights
§18.32 OPs with Respect to Freud Weights
No explicit expressions for the corresponding OPs are available. However, for asymptotic approximations in terms of elementary functions for the OPs, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OPs in the case Q ( x ) = x 4 see Bo and Wong (1999). For asymptotic approximations to OPs that correspond to Freud weights with more general functions Q ( x ) see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).
3: 18.36 Miscellaneous Polynomials
Similar OPs can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)).
§18.36(ii) Sobolev OPs
Sobolev OPs are orthogonal with respect to an inner product involving derivatives. …
§18.36(iii) Multiple OPs
Classes of such polynomials have been found that generalize the classical OPs in the sense that they satisfy second-order matrix differential equations with coefficients independent of the degree. …
4: 18.38 Mathematical Applications
§18.38(i) Classical OPs: Numerical Analysis
Classical OPs play a fundamental role in Gaussian quadrature. …
§18.38(ii) Classical OPs: Other Applications
§18.38(iii) Other OPs
For group-theoretic interpretations of OPs see Vilenkin and Klimyk (1991, 1992, 1993). …
5: 18.1 Notation
x , y

real variables.

OPs

orthogonal polynomials.

q -Hahn Class OPs
Askey–Wilson Class OPs
Other OPs
Classical OPs in Two Variables
6: 18.30 Associated OP’s
§18.30 Associated OPs
Then the polynomials p n ( x , c ) generated in this manner are called corecursive associated OPs. …
7: 18.37 Classical OP’s in Two or More Variables
§18.37 Classical OPs in Two or More Variables
§18.37(ii) OPs on the Triangle
§18.37(iii) OPs Associated with Root Systems
8: 18.2 General Orthogonal Polynomials
If the orthogonality discrete set X is { 0 , 1 , , N } or { 0 , 1 , 2 , } , then the role of the differentiation operator d / d x in the case of classical OPs18.3) is played by Δ x , the forward-difference operator, or by x , the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OPs18.20(i)). … then two special normalizations are: (i) orthonormal OPs: h n = 1 , k n > 0 ; (ii) monic OPs: k n = 1 . … If the OPs are orthonormal, then c n = a n - 1 ( n 1 ). If the OPs are monic, then a n = 1 ( n 0 ). …
9: 18.8 Differential Equations
Table 18.8.1: Classical OPs: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
f ( x ) A ( x ) B ( x ) C ( x ) λ n
10: 18.39 Physical Applications
Classical OPs appear when the time-dependent Schrödinger equation is solved by separation of variables. … For interpretations of zeros of classical OPs as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).