Dixon 3F2(1) sum

(0.005 seconds)

11—20 of 850 matching pages

11: 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).

12: 34.1 Special Notation

… ►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, ►

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},

… ►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient …see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). For other notations for

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

13: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►and the symbol

o\left(1\right)

denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … ►Uniform approximations in terms of Airy functions for the

\mathit{3j}

and

\mathit{6j}

symbols are given in Schulten and Gordon (1975b). For approximations for the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

14: 34.10 Zeros

§34.10 Zeros

►In a

\mathit{3j}

symbol, if the three angular momenta

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

do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the

\mathit{3j}

symbol is zero. Similarly the

\mathit{6j}

symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four

\mathit{3j}

symbols in the summation. …However, the

\mathit{3j}

and

\mathit{6j}

symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

15: 22.9 Cyclic Identities

… ►Throughout this subsection

m

and

p

are positive integers with

1\leq m\leq p

. … ►In this subsection

1\leq m\leq p

and

1\leq n\leq p

. … ►These identities are cyclic in the sense that each of the indices

m, n

in the first product of, for example, the form

s_{m,p}^{(4)}s_{n,p}^{(4)}

are simultaneously permuted in the cyclic order:

m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1

;

n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1

. … ►

§22.9(iii) Typical Identities of Rank 3

… ►

22.9.23

s_{1,3}^{(4)}d_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)% }c_{3,3}^{(4)}c_{1,3}^{(4)}+s_{3,3}^{(4)}d_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).

ⓘ

Symbols:: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

16: 23.9 Laurent and Other Power Series

… ►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), …Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as

1/\wp\left(z\right)\to 0

. … ►where

a_{0,0}=1

a_{m,n}=0

if either

m

n<0

, and …For

a_{m,n}

with

m=0,1,\dots,12

and

n=0,1,\dots,8

, see Abramowitz and Stegun (1964, p. 637).

17: 10.19 Asymptotic Expansions for Large Order

… ►If

\nu\to\infty

through positive real values with

\beta

\left(\in\left(0,\tfrac{1}{2}\pi\right)\right)

fixed, and …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

… ►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. …

18: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D}^{\prime}3,n\right)

denotes the number of partitions of

n

into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …The set

\{2,3,4,\ldots\}

is denoted by

T

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

k

for which

n-(3k^{2}\pm k)\geq 0

. … ►It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). …

19: 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. …

20: 36.6 Scaling Relations

… ►

\text{umbilics: }\mathbf{y}^{(\mathrm{U})}(k)=\left(xk^{2/3},yk^{2/3},zk^{1/3}% \right).

… ►

\text{umbilics: }\beta^{(\mathrm{U})}=\frac{1}{3}.

… ►

\gamma_{z}^{(\mathrm{U})}=\tfrac{1}{3}.

… ►

\text{cuspoids: }\gamma_{K}=\sum\limits_{m=1}^{K}\gamma_{mK}=\dfrac{K(K+3)}{2(% K+2)},

►

\text{umbilics: }\gamma^{(\mathrm{U})}=\sum\limits_{m=1}^{3}\gamma_{m}^{(% \mathrm{U})}=\tfrac{5}{3}.

…