orthogonality properties

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1: 12.16 Mathematical Applications

… ►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. …

2: 18.3 Definitions

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With the property that ${\{p_{n+1}^{\prime }(x)\}}_{n=0}^{\mathrm{\infty }}$ is again a system of OP’s. See §18.9(iii).

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, 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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… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). …

3: 18.19 Hahn Class: Definitions

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Hahn, Krawtchouk, Meixner, and Charlier

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Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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… ►A special case of (18.19.8) is $w^{(1/2)}(x;\pi /2)=\frac{\pi }{\cosh \left(\pi x\right)}$.

4: 3.11 Approximation Techniques

… ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ►The $c_n$ in (3.11.11) can be calculated from (3.11.10), but in general it is more efficient to make use of the orthogonal property (3.11.9). …

5: 14.17 Integrals

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§14.17(iii) Orthogonality Properties

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6: 18.35 Pollaczek Polynomials

… ► … ►Then … ►For type 3 orthogonality (18.35.5) generalizes to …

7: 18.25 Wilson Class: Definitions

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Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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8: 18.34 Bessel Polynomials

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§18.34(ii) Orthogonality

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9: 18.5 Explicit Representations

… ►However, in these circumstances the orthogonality property (18.2.1) disappears. … …

10: 18.27 $q$-Hahn Class

… ►All these systems of OP’s have orthogonality properties of the form …Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. …