# OP’s

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## 1—10 of 23 matching pages

##### 1: 18.40 Methods of Computation

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►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 OP’s of large degree.
►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.
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##### 2: 18.32 OP’s with Respect to Freud Weights

###### §18.32 OP’s with Respect to Freud Weights

… ►No explicit expressions for the corresponding OP’s are available. However, for asymptotic approximations in terms of elementary functions for the OP’s, 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 OP’s in the case $Q(x)={x}^{4}$ see Bo and Wong (1999). ►For asymptotic approximations to OP’s 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

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►Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)).
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###### §18.36(ii) Sobolev OP’s

►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … ►###### §18.36(iii) Multiple OP’s

… ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second-order matrix differential equations with coefficients independent of the degree. …##### 4: 18.38 Mathematical Applications

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

… ►Classical OP’s play a fundamental role in Gaussian quadrature. … ►###### §18.38(ii) Classical OP’s: Other Applications

… ►###### §18.38(iii) Other OP’s

… ►For group-theoretic interpretations of OP’s see Vilenkin and Klimyk (1991, 1992, 1993). …##### 5: 18.1 Notation

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$x,y$ |
real variables. |
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OP’s |
orthogonal polynomials. |

###### $q$-Hahn Class OP’s

… ►###### Askey–Wilson Class OP’s

… ►###### Other OP’s

… ►###### Classical OP’s in Two Variables

…##### 6: 18.30 Associated OP’s

###### §18.30 Associated OP’s

… ►Then the polynomials ${p}_{n}(x,c)$ generated in this manner are called*corecursive associated OP’s*. …

##### 7: 18.37 Classical OP’s in Two or More Variables

###### §18.37 Classical OP’s in Two or More Variables

… ►###### §18.37(ii) OP’s on the Triangle

… ►###### §18.37(iii) OP’s Associated with Root Systems

…##### 8: 18.2 General Orthogonal Polynomials

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►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots},N\}$ or $\{0,1,2,\mathrm{\dots}\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta}}_{x}$, the forward-difference operator, or by ${\nabla}_{x}$, the backward-difference operator; compare §18.1(i).
This happens, for example, with the Hahn class OP’s (§18.20(i)).
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►then two special normalizations are: (i)

*orthonormal OP’s*: ${h}_{n}=1$, ${k}_{n}>0$; (ii)*monic OP’s*: ${k}_{n}=1$. … ►If the OP’s are orthonormal, then ${c}_{n}={a}_{n-1}$ ($n\ge 1$). If the OP’s are monic, then ${a}_{n}=1$ ($n\ge 0$). …##### 9: 18.8 Differential Equations

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