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11: 20 Theta Functions

Chapter 20 Theta Functions

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12: 28.27 Addition Theorems

… ►They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

13: Bibliography G

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A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

Encodings:

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The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Notes:

Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.

External Links:

GAP

Referenced by:

2nd item.

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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14: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

Encodings:

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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.

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Referenced by:

§18.39(iii).

Encodings:

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Discrete ordinate solution of a Fokker-Planck equation in laser physics. Transport Theory Statist. Phys. 15 (1-2), pp. 181–210.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

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External Links:

ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)

Referenced by:

§18.3.

Encodings:

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15: 18.27 $q$-Hahn Class

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§18.27(vii) Discrete $q$-Hermite I and II Polynomials

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Discrete $q$-Hermite I

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Discrete $q$-Hermite II

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18.27.24 $$\sum _{\mathrm{\ell }=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\tilde{h}}_n(c{q}^{\mathrm{\ell }};q){\tilde{h}}_m(c{q}^{\mathrm{\ell }};q)+{\tilde{h}}_n(-c{q}^{\mathrm{\ell }};q){\tilde{h}}_m(-c{q}^{\mathrm{\ell }};q)\right)\frac{q^{\mathrm{\ell }}}{{(-c^2{q}^{2\mathrm{\ell }};q^2)}_{\mathrm{\infty }}}=2\frac{{(q^2,-c^{2}q,-c^{-2}q;q^2)}_{\mathrm{\infty }}}{{(q,-c^2,-c^{-2}{q}^2;q^2)}_{\mathrm{\infty }}}\frac{{(q;q)}_n}{q^{n^2}}{\delta }_{n,m},$$ $c>0$.

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${\tilde{h}}_n(x;q)$: discrete $q$-Hermite II polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.27.E24

Encodings:

pMML, png, TeX

See also:

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

►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

16: Bibliography R

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton, FL.

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External Links:

ISBN 0-8493-0149-1, MathReview, ZentralBlatt

Referenced by:

§26.1, §26.1, §26.11, §26.19, Ch.26, Notations, Notations.

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17: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. … ►The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.

18: Bibliography Y

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

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19: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

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20: Bibliography L

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

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Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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External Links:

ISBN 9780470141731, Document, Link

Referenced by:

§18.38(i).

Encodings:

BibTeX

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