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11: 34.7 Basic Properties: $\mathit{9}j$ Symbol

§34.7 Basic Properties: $\mathit{9}j$ Symbol

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§34.7(ii) Symmetry

►The $\mathit{9}j$ symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent $\mathit{9}j$ symbols. … ►

§34.7(iii) Recursion Relations

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§34.7(iv) Orthogonality

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12: 20.1 Special Notation

… ►The main functions treated in this chapter are the theta functions ${\theta }_j\left(z\right|\tau )={\theta }_j(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. When $\tau$ is fixed the notation is often abbreviated in the literature as ${\theta }_j(z)$, or even as simply ${\theta }_j$, it being then understood that the argument is the primary variable. … ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. … ►Here the symbol $\mathrm{H}$ denotes capital eta. … ►McKean and Moll’s notation: ${\vartheta }_j(z|\tau )={\theta }_j\left(\pi z\right|\tau )$, $j=1,2,3,4$. …

13: 18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

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Recurrence Relation

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§18.28(viii) $q$-Racah Polynomials

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§18.28(x) Limit Relations

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

14: 3.11 Approximation Techniques

… ►They satisfy the recurrence relation … ►Here the single prime on the summation symbol means that the first term is to be halved. … ► … ►Also, in cases where $f(x)$ satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the $c_n$. … ►With $w(x)=1$ and 14-digit computation, we obtain the following rational approximation of type $[3,3]$ to the Bessel function $J_0\left(x\right)$ (§10.2(ii)) on the interval $0\le x\le j_{0,1}$, where $j_{0,1}$ is the first positive zero of $J_0\left(x\right)$: …

15: Software Index

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This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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A cross index of mathematical software in use at NIST.

16: 18.27 $q$-Hahn Class

… ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►

From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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From Little $q$-Laguerre to Laguerre

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17: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

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§35.9.

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

Referenced by:

§16.4(iii), §16.4(iii).

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Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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External Links:

Document

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§34.13.

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

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§34.6.

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The $3$-$j$ and $6$-$j$ Symbols. The Technology Press, MIT, Cambridge, MA.

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External Links:

ISBN 0-262-18004-9

Referenced by:

(34.1.1), §34.1, §34.1, §34.14, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.

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18: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

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§28.32(i).

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

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Document, Code 1, Code 2, Code 3

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8th item.

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Semiclassical approximations to $3j$- and $6j$-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1971–1988.

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External Links:

Document, MathReview (J.-L. Calais)

Referenced by:

§34.8.

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Arbitrary $3n-j$ symbols for $\mathrm{SU}(2)$ . Comput. Phys. Comm. 1 (3), pp. 207–215.

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Notes:

Single-precision Fortran

External Links:

Document, Code

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9th item.

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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19: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►

§18.30(i) Associated Jacobi Polynomials

… ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. … ►More generally, the $k$th corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the $c=k$ recurrence of (18.30.27)) are related to the $(k+1)$st monic associated polynomials by …See Ismail (2009, p. 46 ), where the $k$th corecursive polynomial is also related to an appropriate continued fraction, given here as its $n$th convergent, …

20: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

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§23.21(ii).

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The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau

External Links:

ISBN 978-1-611976-38-0, MathReview Entry

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§18.40(ii).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:

ISBN 978-3-540-04218-1

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§34.14.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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