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21: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:

Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The Gaussian Function in Calculations of Statistical Mechanics and Quantum Mechanics. In Methods in Computational Physics: Advances in Research and Applications, B. Alder, S. Fernbach, and M. Rotenberg (Eds.), Vol. 2, pp. 1–45.

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Referenced by:

§8.24(i), §8.24(iii).

Encodings:

BibTeX

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Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

MathReview (P. C. Hammer), ZentralBlatt

Referenced by:

§3.5(v).

Encodings:

BibTeX

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22: Bibliography G

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How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

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External Links:

ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§18.40(ii), §3.5(v), §9.17(iii).

Encodings:

BibTeX

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Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

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External Links:

ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt

Referenced by:

§10.74(iii), §9.17(iii).

Encodings:

BibTeX

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23: 18.38 Mathematical Applications

… ►Classical OP’s play a fundamental role in Gaussian quadrature. … ►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. …

24: Bibliography R

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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25: Bibliography H

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Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

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External Links:

ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§15.20(iii).

Encodings:

BibTeX

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26: 18.39 Applications in the Physical Sciences

… ►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. … ►

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

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Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
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Half-Freud Gaussian	$\exp \left(-{(x-x_0)}^2\right)$	$[0,\mathrm{\infty })$	$G_n(x)$	Fokker–Planck DVRe
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… ►Full expressions for both $A_{x_i,l}$ and $B_l(x)$ are given in Yamani and Reinhardt (1975) and it is seen that ${|A_{x_i,l}/B_l(x_i)|}^2$ = $w_i^N/w^{\mathrm{CP}}(x_i)$ where $w_i^N$ is the Gaussian-Pollaczek quadrature weight at $x=x_i$, and $w^{\mathrm{CP}}(x_i)$ is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …

27: 18.36 Miscellaneous Polynomials

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. …

28: 26.10 Integer Partitions: Other Restrictions

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26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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29: 18.27 $q$-Hahn Class

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18.27.4 $$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $y$: real variable, $N$: positive integer, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Notes:

This is Ismail (2009, (18.5.2)) but the presentation has been simplified.

Referenced by:

§18.27(ii), Erratum (V1.2.0) for Equation (18.27.4)

Permalink:

http://dlmf.nist.gov/18.27.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.4_1 $$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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30: Bibliography B

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Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10 (12), pp. 2061–2081.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§36.6.

Encodings:

BibTeX

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