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11: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►Examples are provided by: … ►

§34.5(vi) Sums

… ►Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the

\mathit{6j}

symbol. … ►For other sums see Ginocchio (1991).

12: 34.3 Basic Properties: $\mathit{3j}$ Symbol

§34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►

§34.3(ii) Symmetry

… ►

§34.3(iv) Orthogonality

… ►

§34.3(vi) Sums

►For sums of products of

\mathit{3j}

symbols, see Varshalovich et al. (1988, pp. 259–262). …

13: 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: … ►Except in degenerate cases the combination of the triangle inequalities for the four

\mathit{3j}

symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths

j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}

; see Figure 34.4.1. … ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

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

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

14: 5.16 Sums

§5.16 Sums

… ►

5.16.2

\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right)=\zeta\left(3\right)=-\frac% {1}{2}\psi''\left(1\right).

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.16 and Ch.5

►For further sums involving the psi function see Hansen (1975, pp. 360–367). For sums of gamma functions see Andrews et al. (1999, Chapters 2 and 3) and §§15.2(i), 16.2. ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

15: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for

\mathit{3j}

and

\mathit{6j}

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{9j}

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). …

16: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

17: 23.9 Laurent and Other Power Series

… ►

23.9.1

c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},

n=2,3,4,\dots

ⓘ

Symbols:: $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $n$ : integer and $c_{n}$
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/23.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►

c_{3}=\frac{1}{28}g_{3},

►

23.9.5

c_{n}=\frac{3}{(2n+1)(n-3)}\sum_{m=2}^{n-2}c_{m}c_{n-m},

n\geq 4

ⓘ

Symbols:: $n$ : integer, $m$ : integer and $c_{n}$
A&S Ref:: 18.5.3
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

►Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). ►For

j=1,2,3

, and with

e_{j}

as in §23.3(i), …

18: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

19: 10.19 Asymptotic Expansions for Large Order

… ►In these expansions

U_{k}(p)

and

V_{k}(p)

are the polynomials in

p

of degree

3k

defined in §10.41(ii). … ►with sectors of validity

-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}\nu\leq\tfrac{3}{2}\pi-\delta

. … ►

J_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)\sim-\frac{2^{\frac{2}{3}}}{\nu^{% \frac{2}{3}}}\operatorname{Ai}'\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}% \operatorname{Ai}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(% a)}{\nu^{2k/3}},

|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta

►

Y_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)\sim\frac{2^{\frac{2}{3}}}{\nu^{% \frac{2}{3}}}\operatorname{Bi}'\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{R_{k}(a)}{\nu^{2k/3}}-\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}% \operatorname{Bi}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(% a)}{\nu^{2k/3}},

|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta

… ►with sectors of validity

-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}\nu\leq\tfrac{3}{2}\pi-\delta

and

-\tfrac{3}{2}\pi+\delta\leq\operatorname{ph}\nu\leq\tfrac{1}{2}\pi-\delta

, respectively. …

20: 23.3 Differential Equations

… ►and are denoted by

e_{1},e_{2},e_{3}

. … ►Let

{g_{2}}^{3}\neq 27{g_{3}}^{2}

, or equivalently

\Delta

be nonzero, or

e_{1},e_{2},e_{3}

be distinct. …Similarly for

\zeta\left(z;g_{2},g_{3}\right)

and

\sigma\left(z;g_{2},g_{3}\right)

. As functions of

g_{2}

and

g_{3}

\wp\left(z;g_{2},g_{3}\right)

and

\zeta\left(z;g_{2},g_{3}\right)

are meromorphic and

\sigma\left(z;g_{2},g_{3}\right)

is entire. ►Conversely,

g_{2}

g_{3}

, and the set

\{e_{1},e_{2},e_{3}\}

are determined uniquely by the lattice

\mathbb{L}

independently of the choice of generators. …