finite sum of 6j symbols

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11: 18.38 Mathematical Applications

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$3j$ and $6j$ Symbols

… ► ►The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${}_4{}^{}F_3^{}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … …

12: 24.4 Basic Properties

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§24.4(iii) Sums of Powers

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24.4.11 $$\sum _{\begin{array}{c}k=1\\ \left(k,m\right)=1\end{array}}^m{k}^n=\frac{1}{n+1}\sum _{j=1}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{j}\right)\left(\prod _{p|m}(1-p^{n-j})B_{n+1-j}\right)m^j.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\left(m,n\right)$: greatest common divisor (gcd), $j$: integer, $k$: integer, $m$: integer, $n$: integer and $p$: prime

Referenced by:

§24.4(iii)

Permalink:

http://dlmf.nist.gov/24.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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§24.4(iv) Finite Expansions

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24.4.19 $$E_n\left(mx\right)=-\frac{2m^n}{n+1}\sum _{k=0}^{m-1}{(-1)}^k{B}_{n+1}\left(x+\frac{k}{m}\right),$$ $m=2,4,6,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.10

Permalink:

http://dlmf.nist.gov/24.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

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§24.4(viii) Symbolic Operations

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13: 2.4 Contour Integrals

… ►If, in addition, the corresponding integrals with $Q$ and $F$ replaced by their derivatives $Q^{(j)}$ and $F^{(j)}$, $j=1,2,\mathrm{\dots },m$, converge uniformly, then by repeated integrations by parts … ►in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. … ►

In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\lambda$: constant, $a$: left endpoint, $p(t)$: analytic function, $q(t)$: analytic function, $p_s$: coefficients, $q_s$: coefficients and $\mu$: positive constant

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E11

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

with $\mathrm{\Re }\lambda >0$, $\mu >0$, $p_0\ne 0$, and the branches of ${(t-a)}^{\lambda }$ and ${(t-a)}^{\mu }$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega$ as $t\to a$ along $𝒫$.

… ►with $a$ and $b$ chosen so that the zeros of $\partial p(\alpha ,t)/\partial t$ correspond to the zeros $w_1(\alpha ),w_2(\alpha )$, say, of the quadratic $w^2+2aw+b$. … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …