product representation

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21: 10.22 Integrals

… ►See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. …

22: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

►The

\mathit{6j}

symbol is defined by the following double sum of products of

\mathit{3j}

symbols: … ►

34.4.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: Erratum (V1.0.7) for Equation (34.4.2)
Permalink:: http://dlmf.nist.gov/34.4.E2
Encodings:: pMML, png, TeX
Errata (effective with 1.0.7):: Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.
Reported 2012-12-31 by Yu Lin
See also:: Annotations for §34.4 and Ch.34

… ►

34.4.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

►where

{{}_{4}F_{3}}

is defined as in §16.2. …

… ►NIST makes no warranties or representations as to the correctness, accuracy, or reliability of the DLMF. … ►Certain products, commercial and otherwise, are mentioned in the DLMF. … ►NIST expressly does not endorse or recommend any specific product or service. …

24: Bibliography S

… ►

B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

ⓘ

Notes:: First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
External Links:: ISBN 0-387-95067-2, MathReview, ZentralBlatt
Referenced by:: §26.19.
Encodings:: BibTeX

… ►

B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

ⓘ

Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt
Referenced by:: §29.17(i), §29.8.
Encodings:: BibTeX

… ►

H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… ►

N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

ⓘ

External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

…

25: 5.14 Multidimensional Integrals

… ►

5.14.7

\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}|e^{i\theta_{j% }}-e^{i\theta_{k}}|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},

\Re b>-1/n

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $b$ : real or complex variable
Referenced by:: §5.14, §5.20
Permalink:: http://dlmf.nist.gov/5.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

26: Bibliography G

… ►

A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.18.
Encodings:: BibTeX

… ►

E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

ⓘ

External Links:: ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.6(iii).
Encodings:: BibTeX

… ►

H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

ⓘ

External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

K. I. Gross and R. A. Kunze (1976) Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

ⓘ

External Links:: Document, MathReview (E. A. Thieleker), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

… ►

E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

27: 16.11 Asymptotic Expansions

… ►

16.11.2

H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.

ⓘ

Defines:: $H_{p,q}(z)$ : formal infinite series (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.11(ii), §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(i), §16.11 and Ch.16

… ►

16.11.5

e_{k,m}=\sum_{j=1}^{q+1}{\left(1-\nu-\kappa b_{j}+m\right)_{\kappa+k-m}}\left(% {\textstyle\ifrac{\prod\limits_{\ell=1}^{p}(a_{\ell}-b_{j})}{\prod\limits_{% \begin{subarray}{c}\ell=1\\ \ell\neq j\end{subarray}}^{q+1}(b_{\ell}-b_{j})}}\right),

ⓘ

Defines:: $e_{k,m}$ (locally)
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $\kappa$ and $\nu$
Referenced by:: §16.11, Erratum (V1.1.9) for Section 16.11(i)
Permalink:: http://dlmf.nist.gov/16.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(i), §16.11 and Ch.16

… ►Explicit representations for the coefficients

c_{k}

are given in Volkmer (2023). … ►

16.11.6

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),

|\operatorname{ph}\left(-z\right)|\leq\pi

;

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $H_{p,q}(z)$ : formal infinite series
Permalink:: http://dlmf.nist.gov/16.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►Explicit representations for the coefficients

c_{k}

are given in Volkmer and Wood (2014). …

28: 5.19 Mathematical Applications

… ►Many special functions

f(z)

can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of

z

, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of

f(z)

for large

|z|

, or small

|z|

, can be obtained complete with an integral representation of the error term. …

29: Errata

… ►

Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… ►

Equation (20.4.2)

20.4.2

\theta_{1}'\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)^{3}=2q^{1/4}{\left(q^{2};q^{2}\right)_{\infty}}^{3}

The representation in terms of ${\left(q^{2};q^{2}\right)_{\infty}}^{3}$ was added to this equation.

… ►

Equation (27.14.2)

27.14.2

\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})=\left(x;x\right)_{% \infty},

|x|<1

The representation in terms of $\left(x;x\right)_{\infty}$ was added to this equation.

…

30: 11.9 Lommel Functions

… ►

11.9.4

a_{k}(\mu,\nu)=\prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)=4^{k}{\left(% \frac{\mu-\nu+1}{2}\right)_{k}}{\left(\frac{\mu+\nu+1}{2}\right)_{k}},

k=0,1,2,\dots

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\nu$ : real or complex order, $k$ : nonnegative integer and $a_{k}(\mu,\nu)$ : expansion function
Referenced by:: §11.9(iii), Erratum (V1.1.3) for Additions
Permalink:: http://dlmf.nist.gov/11.9.E4
Encodings:: pMML, png, TeX
Addition (effective with 1.1.3):: An alternative Pochhammer symbol representation was added.
See also:: Annotations for §11.9(i), §11.9 and Ch.11

…