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##### 1: 19.2 Definitions
where $p_{j}$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. … Here $a,b,p$ are real parameters, and $k_{c}$ and $x$ are real or complex variables, with $p\neq 0$, $k_{c}\neq 0$. … If $1, then $k_{c}$ is pure imaginary. …
###### §19.2(iv) A Related Function: $R_{C}\left(x,y\right)$
For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15). …
##### 2: 34.6 Definition: $\mathit{9j}$ Symbol
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
34.6.2 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
##### 3: 34.7 Basic Properties: $\mathit{9j}$ Symbol
34.7.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.$
34.7.2 $\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.$
34.7.3 $\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.$
34.7.4 $\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\*\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}.$
34.7.5 $\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.$
##### 4: 34.1 Special Notation
$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$
$\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.$
An often used alternative to the $\mathit{3j}$ symbol is the Clebsch–Gordan coefficient
34.1.1 $\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};$
see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). …
##### 5: 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. … It follows that $p_{k}\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. $p_{k}\left(\leq m,n\right)$ is the number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$. …
##### 6: 27.2 Functions
where $p_{1},p_{2},\dots,p_{\nu\left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_{r}$ is positive, and $\nu\left(n\right)$ is the number of distinct primes dividing $n$. … Note that $\sigma_{0}\left(n\right)=d\left(n\right)$. …Note that $J_{1}\left(n\right)=\phi\left(n\right)$. In the following examples, $a_{1},\dots,a_{\nu\left(n\right)}$ are the exponents in the factorization of $n$ in (27.2.1). … Table 27.2.1 lists the first 100 prime numbers $p_{n}$. …
##### 7: 26.16 Multiset Permutations
Let $S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}$ be the multiset that has $a_{j}$ copies of $j$, $1\leq j\leq n$. $\mathfrak{S}_{S}$ denotes the set of permutations of $S$ for all distinct orderings of the $a_{1}+a_{2}+\cdots+a_{n}$ integers. The number of elements in $\mathfrak{S}_{S}$ is the multinomial coefficient (§26.4) $\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}$. … The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with $S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}$ we have …
##### 8: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
$\genfrac{(}{)}{0.0pt}{}{n}{n_{1},n_{2},\ldots,n_{k}}$ is the number of ways of placing $n=n_{1}+n_{2}+\cdots+n_{k}$ distinct objects into $k$ labeled boxes so that there are $n_{j}$ objects in the $j$th box. … These are given by the following equations in which $a_{1},a_{2},\ldots,a_{n}$ are nonnegative integers such that …$M_{1}$ is the multinominal coefficient (26.4.2): …For each $n$ all possible values of $a_{1},a_{2},\ldots,a_{n}$ are covered. … where the summation is over all nonnegative integers $n_{1},n_{2},\ldots,n_{k}$ such that $n_{1}+n_{2}+\cdots+n_{k}=n$. …
##### 9: 1.3 Determinants
The cofactor $A_{jk}$ of $a_{jk}$ is … For real-valued $a_{jk}$, … where $\omega_{1},\omega_{2},\dots,\omega_{n}$ are the $n$th roots of unity (1.11.21). … If $D_{n}[a_{j,k}]$ tends to a limit $L$ as $n\to\infty$, then we say that the infinite determinant $D_{\infty}[a_{j,k}]$ converges and $D_{\infty}[a_{j,k}]=L$. … Here $\delta_{j,k}$ is the Kronecker delta. …
##### 10: 34.3 Basic Properties: $\mathit{3j}$ Symbol
When any one of $j_{1},j_{2},j_{3}$ is equal to $0,\tfrac{1}{2}$, or $1$, the $\mathit{3j}$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_{1},j_{2},j_{3}$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). … Even permutations of columns of a $\mathit{3j}$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{j_{1}+j_{2}+j_{3}}$, for example,
34.3.8 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix},$
For the polynomials $P_{l}$ see §18.3, and for the function $Y_{{l},{m}}$ see §14.30. …