q-Pochhammer symbol

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11—20 of 210 matching pages

11: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …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).

12: 34.1 Special Notation

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

►The main functions treated in this chapter are the Wigner

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

symbols, respectively, … ►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient ►

34.1.1

\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};

ⓘ

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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

… ►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 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

…

14: 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.

15: Peter Paule

… … ► 1958 in Ried im Innkreis, Austria) is Professor of Mathematics (successor to Bruno Buchberger), Director of the Research Institute for Symbolic Computation (RISC), and Director of the Doctoral Program on Computational Mathematics at the Johannes Kepler University, Linz, Austria. … ►He is on the editorial boards for the Journal of Symbolic Computation and The Ramanujan Journal, and is Managing Editor of Annals of Combinatorics. He is also Editor-in-Chief of the Springer book series Texts and Monographs in Symbolic Computation. …

16: 15.11 Riemann’s Differential Equation

… ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$ -symbol: ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►

15.11.4

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►Symbolically: ►

15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

…

17: 16.24 Physical Applications

… ►

§16.24(iii) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ Symbols

►The

\mathit{3j}

symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner

\mathit{6j}

symbols. …Lastly, special cases of the

\mathit{9j}

symbols are

{{}_{5}F_{4}}

functions with unit argument. …

18: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

§34.7(ii) Symmetry

… ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

… ►It constitutes an addition theorem for the

\mathit{9j}

symbol. …

19: 27.1 Special Notation

… ► ►►►►

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$(n\|P)$	Jacobi symbol; see §27.9.
$(n\|p)$	Legendre symbol; see §27.9.

20: 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

… ►

q-Pochhammer symbol

11—20 of 210 matching pages

11: 34.9 Graphical Method

§34.9 Graphical Method

12: 34.1 Special Notation

13: 34 3j, 6j, 9j Symbols

Chapter 34 3 ⁢ j , 6 ⁢ j , 9 ⁢ j Symbols

14: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

15: Peter Paule

16: 15.11 Riemann’s Differential Equation

17: 16.24 Physical Applications

§16.24(iii) 3 ⁢ j , 6 ⁢ j , and 9 ⁢ j Symbols

18: 34.7 Basic Properties: 9 ⁢ j Symbol

§34.7 Basic Properties: 9 ⁢ j Symbol

§34.7(ii) Symmetry

§34.7(iv) Orthogonality

§34.7(vi) Sums

19: 27.1 Special Notation

20: 34.3 Basic Properties: 3 ⁢ j Symbol

§34.3 Basic Properties: 3 ⁢ j Symbol

§34.3(ii) Symmetry

§34.3(iv) Orthogonality

§34.3(vi) Sums

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

§16.24(iii) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ Symbols

18: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

20: 34.3 Basic Properties: $\mathit{3j}$ Symbol

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