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1: 34.2 Definition: $\mathit{3}j$ Symbol

§34.2 Definition: $\mathit{3}j$ Symbol

►The quantities $j_1,j_2,j_3$ in the $\mathit{3}j$ symbol are called angular momenta. …They therefore satisfy the triangle conditions …where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by …

2: Simon Ruijsenaars

… ► 1948 in Hillegom, The Netherlands) is a Professor of Mathematical Physics at the University of Leeds. … ►

28 Mathieu Functions and Hill’s Equation

3: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

…

4: 31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

5: Staff

… ►

Ronald F. Boisvert, Editor at Large, NIST

… ►

Gerhard Wolf, University of Duisberg-Essen, Chap. 28

… ►

Diego Dominici, State University of New York at New Paltz, for Chaps. 9, 10 (deceased)

… ►

Amparo Gil, Universidad de Cantabria, for Chap. 3

… ►

Simon Ruijsenaars, University of Leeds, for Chaps. 5, 28

…

6: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

► ►►►►

$m$	$n$
	…
3	1	3	3	1
…
8	1	8	28	56	70	56	28	8	1
…

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

► ►►►►

$m$	$n$
	…
2	1	3	6	10	15	21	28	36	45
…
6	1	7	28	84	210	462	924	1716	3003
…

►

… ►

26.3.11 $$\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)=\frac{2^n(2n-1)(2n-3)\mathrm{\cdots }3\cdot 1}{n!}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$) and $n$: nonnegative integer

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iv), §26.3 and Ch.26

…

7: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$. … ► $p_k(\le m,n)$ is the number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$. …

8: Gerhard Wolf

… ►

28 Mathieu Functions and Hill’s Equation

…

9: Bibliography Y

… ►

On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

Encodings:

BibTeX

… ►

Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

… ►

Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

BibTeX

…

10: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …