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1: 34.13 Methods of Computation

… ►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For $\mathit{9}j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

2: 36 Integrals with Coalescing Saddles

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3: 15.17 Mathematical Applications

… ►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

4: 27.16 Cryptography

… ►With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. …

5: 31.9 Orthogonality

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§31.9(i) Single Orthogonality

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6: Notices

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In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

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7: 3.1 Arithmetics and Error Measures

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IEEE Standard

… ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ►

Figure 3.1.1: Floating-point arithmetic. Representation of data in the binary interchange formats for binary32, binary64 and binary128 (previously single, double and quad precision).

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8: Guide to Searching the DLMF

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Table 3: A sample of recognized symbols

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Symbols	Comments
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’ (single quote)	As a prime, which can stand for the derivative
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Single-letter terms

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9: 26.13 Permutations: Cycle Notation

… ►A transposition is a permutation that consists of a single cycle of length two. …A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): …

10: Bibliography

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Algorithm 609. A portable FORTRAN subroutine for the Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 480–493.

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Notes:

Single-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 609; package TOMS), MathReview (Marietta J. Tretter), ZentralBlatt

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1st item.

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BibTeX

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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function. ACM Trans. Math. Software 9 (4), pp. 494–502.

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Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 610; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12 (3), pp. 265–273.

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Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, Remark #1. Remark #2. GAMS (module 644; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item, 1st item, 2nd item, 1st item.

Encodings:

BibTeX

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Algorithm 556: Exponential integrals. ACM Trans. Math. Software 6 (3), pp. 420–428.

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Notes:

Single-precision Fortran, maximum accuracy 14S. For a Remark see same journal v. 9 (1983), no. 4, p. 525.

External Links:

ISSN 0098-3500, Document, Remark,GAMS (module 556; package TOMS), ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

Encodings:

BibTeX

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