discrete weights

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1: 18.39 Applications in the Physical Sciences

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. …For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: … ►

$w_{x_k}^{\mathrm{CP}}={\displaystyle \frac{(l+1+\frac{2Z}{s}){\rho }^{2k-1}{\left(1-{\rho }^2\right)}^{2l+3}{\left(2l+2\right)}_k}{2(k+l+1)k!}}.$

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2: 18.30 Associated OP’s

… ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). …

3: 18.19 Hahn Class: Definitions

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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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4: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. …

5: 18.25 Wilson Class: Definitions

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§18.25(iii) Weights and Normalizations: Discrete Cases

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6: 18.38 Mathematical Applications

… ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …

7: 3.11 Approximation Techniques

… ►Now suppose that $X_{k\mathrm{\ell }}=0$ when $k\ne \mathrm{\ell }$, that is, the functions ${\varphi }_k(x)$ are orthogonal with respect to weighted summation on the discrete set $x_1,x_2,\mathrm{\dots },x_J$. …

8: 18.35 Pollaczek Polynomials

… ►where, depending on $a,b,\lambda$, $D$ is a discrete subset of $ℝ$ and the $w_{\zeta }^{(\lambda )}(a,b)$ are certain weights. …

9: 18.3 Definitions

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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). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). … ►For $-1-\beta >\alpha >-1$ a finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ is orthogonal on $(1,\mathrm{\infty })$ with weight function $w(x)={(x-1)}^{\alpha }{(x+1)}^{\beta }$. …

10: Bibliography Z

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A proof of Andrews’ $q$-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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External Links:

ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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