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31: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of

{{}_{p}F_{q}}

functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of

{{}_{q+1}F_{q}}

functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

32: Bibliography F

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie (1962) An Index of Mathematical Tables. Vols. I, II. 2nd edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, MA.

ⓘ

Notes:: Table errata: Math. Comp. v.27 (1973), p. 1009.
External Links:: MathReview
Referenced by:: §10.75(i), §11.14(i), §12.19, §13.30, §14.33, §18.41(iii), §19.37(i), §20.15, §22.21, §23.23, §24.20, 4th item, §28.35(iv), §30.17, §33.24, §4.23(viii), §4.46, §5.22(i), §6.19(i), §7.23(i), §8.26(i), §9.18(i).
Encodings:: BibTeX

…

33: 24.2 Definitions and Generating Functions

… ►

B_{2n+1}=0

►

(-1)^{n+1}B_{2n}>0

n=1,2,\dots

… ►

E_{2n+1}=0

…

34: 4.13 Lambert $W$ -Function

… ►

4.13.10

W_{k}\left(z\right)\sim\xi_{k}-\ln\xi_{k}+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{% \xi_{k}^{n}}\sum_{m=1}^{n}\genfrac{[}{]}{0.0pt}{}{n}{n-m+1}\frac{\left(-\ln\xi% _{k}\right)^{m}}{m!},

ⓘ

Symbols:: $W\left(\NVar{z}\right)$ : Lambert $W$ -function, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Stirling cycle number of the first kind, $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : integer, $m$ : integer, $n$ : integer and $z$ : complex variable
Notes:: See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).
Referenced by:: §4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion
Permalink:: http://dlmf.nist.gov/4.13.E10
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: Originally we gave only the first three terms of the infinite series.
See also:: Annotations for §4.13 and Ch.4

… ►

4.13.11

W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim-\eta-\ln\eta+\sum_{n=1}^{\infty}% \frac{1}{\eta^{n}}\sum_{m=1}^{n}\genfrac{[}{]}{0.0pt}{}{n}{n-m+1}\frac{\left(-% \ln\eta\right)^{m}}{m!},

ⓘ

Symbols:: $W\left(\NVar{z}\right)$ : Lambert $W$ -function, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Stirling cycle number of the first kind, $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : integer, $n$ : integer, $x$ : real variable and $\eta$
Notes:: See Corless et al. (1996, p. 350).
Referenced by:: §4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion
Permalink:: http://dlmf.nist.gov/4.13.E11
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: Originally we gave only the first three terms of the infinite series.
See also:: Annotations for §4.13 and Ch.4

…

35: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

1\leq j\leq m

;

\|\mathbf{T}\|<1

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7 and Ch.35

… ►

35.7.8

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}}\*{{}_{2}F_{1}}\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}% -\mathbf{T}\right),

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

;

{\frac{1}{2}}(j+1)-a\in\mathbb{N}

for some

j=1,\ldots,m

;

{\frac{1}{2}}(j+1)-c\notin\mathbb{N}

and

c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}

for all

j=1,\ldots,m

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(ii), Erratum (V1.0.26) for Equation (35.7.8)
Permalink:: http://dlmf.nist.gov/35.7.E8
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.26):: Originally the constraint was written as $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ . The constraint has been replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ ; for details see https://arxiv.org/abs/2002.05248.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

…

36: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Let

p

and

q

be nonnegative integers;

a_{1},\dots,a_{p}\in\mathbb{C}

;

b_{1},\dots,b_{q}\in\mathbb{C}

;

-b_{j}+\tfrac{1}{2}(k+1)\notin\mathbb{N}

1\leq j\leq q

1\leq k\leq m

. … ►If

-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}

for some

j, k

satisfying

1\leq j\leq p

1\leq k\leq m

, then the series expansion (35.8.1) terminates. …

37: 24.14 Sums

… ►

24.14.4

\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

24.14.8

\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}B_{2j}B_{2k}B_{2\ell}=(n-1)(2n-1)B_{2n}+n(% n-\tfrac{1}{2})B_{2n-2},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer and $n$ : integer
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

… ►

24.14.11

\det[B_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{6}\Bigg{/}\left(% \prod_{k=1}^{2n+1}k!\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12

\det[E_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

…

38: Bibliography G

… ►

W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

ⓘ

External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.25(iv), §8.25(v).
Encodings:: BibTeX

… ►

V. V. Golubev (1960) Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..

ⓘ

Notes:: Published for the National Science Foundation by the Israel Program for Scientific Translations, 1960.
External Links:: MathReview, ZentralBlatt
Referenced by:: §22.19(iv).
Encodings:: BibTeX

…

39: Bibliography L

… ►

Lord Kelvin (1891) Popular Lectures and Addresses. Vol. 3, pp. 481–488.

ⓘ

Referenced by:: §36.13.
Encodings:: BibTeX

… ►

D. W. Lozier (1980) Numerical Solution of Linear Difference Equations. NBSIR Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.

ⓘ

Notes:: (Available from National Technical Information Service, Springfield, VA 22161.)
Referenced by:: §16.25, §3.6(vii).
Encodings:: BibTeX

…

40: Software Index

… ►

Research Software.

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.

►

Open Source Collections and Systems.

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.

…