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##### 1: 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 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
##### 2: 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 $\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.$
##### 3: 28.16 Asymptotic Expansions for Large $q$
Let $s=2m+1$, $m=0,1,2,\dots$, and $\nu$ be fixed with $m<\nu. …
28.16.1 $\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$
##### 4: 13.28 Physical Applications
The reduced wave equation $\nabla^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi\eta}\cos\phi$, $y=2\sqrt{\xi\eta}\sin\phi$, $z=\xi-\eta$, can be solved via separation of variables $w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}$, where
$f_{1}(\xi)=\xi^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(1)}(2\mathrm{i}k\xi)$ ,
$f_{2}(\eta)=\eta^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(2)}(-2\mathrm{i}k\eta)$ ,
and $V^{(j)}_{\kappa,\mu}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). … See Chapter 33. …
##### 5: 29.7 Asymptotic Expansions
$p=2m+1,$
29.7.4 $\tau_{1}=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).$
The same Poincaré expansion holds for $b^{m+1}_{\nu}\left(k^{2}\right)$, since …
29.7.7 $\tau_{3}=\frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})% ^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173)),$
In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$ and $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$. …
##### 6: 34.1 Special Notation
 $2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$ nonnegative integers. …
$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$
$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$
An often used alternative to the $\mathit{3j}$ symbol is the Clebsch–Gordan coefficient
34.1.1 $\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};$
##### 7: 34.14 Tables
33–36. Tables of $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 17/2$ are given in Appel (1968) to 6D. …
##### 8: 30.9 Asymptotic Approximations and Expansions
###### §30.9(i) Prolate Spheroidal Wave Functions
As $\gamma^{2}\to+\infty$, with $q=2(n-m)+1$, … The asymptotic behavior of $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and $a^{m}_{n,k}(\gamma^{2})$ as $n\to\infty$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ and $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ as $x\to\pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of $\lambda^{m}_{n}\left(\gamma^{2}\right)$ for complex $\gamma^{2}$ and large $|\lambda^{m}_{n}\left(\gamma^{2}\right)|$ is investigated in Hunter and Guerrieri (1982). …
##### 9: 28.11 Expansions in Series of Mathieu Functions
Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). …See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33). …
28.11.3 $1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\operatorname{ce}_{2n}\left(z,q\right),$
28.11.4 $\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\operatorname{ce}_{2n}\left(z,q% \right),$ $m\neq 0$,
28.11.7 $\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\operatorname{se}_{2n+2}\left% (z,q\right).$
##### 10: 26.2 Basic Definitions
Thus $231$ is the permutation $\sigma(1)=2$, $\sigma(2)=3$, $\sigma(3)=1$. … Here $\sigma(1)=2,\sigma(2)=5$, and $\sigma(5)=1$. … A lattice path is a directed path on the plane integer lattice $\{0,1,2,\ldots\}\times\{0,1,2,\ldots\}$. … As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …