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##### 1: 19.2 Definitions
Assume $1-{\sin}^{2}\phi\in\mathbb{C}\setminus(-\infty,0]$ and $1-k^{2}{\sin}^{2}\phi\in\mathbb{C}\setminus(-\infty,0]$, except that one of them may be 0, and $1-\alpha^{2}{\sin}^{2}\phi\in\mathbb{C}\setminus\{0\}$. … The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\operatorname{ph}\left(1-k^{2}\right)|\leq\pi$, that is, the branch-cuts are $(-\infty,-1]\cup[1,+\infty)$. … Let $k^{\prime}=\sqrt{1-k^{2}}$. … If $1, then $k_{c}$ is pure imaginary. …
##### 2: 10.55 Continued Fractions
For continued fractions for $\mathsf{j}_{n+1}\left(z\right)/\mathsf{j}_{n}\left(z\right)$ and ${\mathsf{i}^{(1)}_{n+1}}\left(z\right)/{\mathsf{i}^{(1)}_{n}}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
##### 3: 4.17 Special Values and Limits
4.17.1 $\lim_{z\to 0}\frac{\sin z}{z}=1,$
4.17.2 $\lim_{z\to 0}\frac{\tan z}{z}=1.$
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 4: 34.5 Basic Properties: $\mathit{6j}$ Symbol
If any lower argument in a $\mathit{6j}$ symbol is $0$, $\tfrac{1}{2}$, or $1$, then the $\mathit{6j}$ symbol has a simple algebraic form. …
34.5.4 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1% )}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$
34.5.11 ${(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$
34.5.18 $\sum_{j}(-1)^{j_{1}+j_{2}+j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{2}&j_{1}&j^{\prime}\end{Bmatrix}=\sqrt{(2j_{1}+1)(2j_{2}+1)}\,\delta_{j^{% \prime},0},$
34.5.22 $\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}=\frac{1}{j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}\left(\frac{% (2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}},$ $j_{2}.
##### 5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
$M_{1}$ is the multinominal coefficient (26.4.2): …$M_{2}$ is the number of permutations of $\{1,2,\ldots,n\}$ with $a_{1}$ cycles of length 1, $a_{2}$ cycles of length 2, $\dotsc$, and $a_{n}$ cycles of length $n$: …$M_{3}$ is the number of set partitions of $\{1,2,\ldots,n\}$ with $a_{1}$ subsets of size 1, $a_{2}$ subsets of size 2, $\dotsc$, and $a_{n}$ subsets of size $n$: …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 $n_{1}+n_{2}+\cdots+n_{k}=n$. …
##### 6: 18.31 Bernstein–Szegő Polynomials
Let $\rho(x)$ be a polynomial of degree $\ell$ and positive when $-1\leq x\leq 1$. The Bernstein–Szegő polynomials $\{p_{n}(x)\}$, $n=0,1,\dots$, are orthogonal on $(-1,1)$ with respect to three types of weight function: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. In consequence, $p_{n}(\cos\theta)$ can be given explicitly in terms of $\rho(\cos\theta)$ and sines and cosines, provided that $\ell<2n$ in the first case, $\ell<2n+2$ in the second case, and $\ell<2n+1$ in the third case. …
##### 7: 24.20 Tables
Abramowitz and Stegun (1964, Chapter 23) includes exact values of $\sum_{k=1}^{m}k^{n}$, $m=1(1)100$, $n=1(1)10$; $\sum_{k=1}^{\infty}k^{-n}$, $\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}$, $\sum_{k=0}^{\infty}(2k+1)^{-n}$, $n=1,2,\dotsc$, 20D; $\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}$, $n=1,2,\dotsc$, 18D. Wagstaff (1978) gives complete prime factorizations of $N_{n}$ and $E_{n}$ for $n=20(2)60$ and $n=8(2)42$, respectively. … For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).
##### 8: 10.10 Continued Fractions
Assume $J_{\nu-1}\left(z\right)\neq 0$. …
10.10.1 $\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}-% \cfrac{1}{2(\nu+1)z^{-1}-\cfrac{1}{2(\nu+2)z^{-1}-\cdots}}},$ $z\neq 0$,
10.10.2 $\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{\tfrac{1}{2}z/\nu% }{1-\cfrac{\tfrac{1}{4}z^{2}/(\nu(\nu+1))}{1-\cfrac{\tfrac{1}{4}z^{2}/((\nu+1)% (\nu+2))}{1-\cdots}}},$ $\nu\neq 0,-1,-2,\dotsc$.
##### 9: 10.33 Continued Fractions
Assume $I_{\nu-1}\left(z\right)\neq 0$. …
10.33.1 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,$ $z\neq 0$,
10.33.2 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,$ $\nu\neq 0,-1,-2,\dotsc$.
##### 10: 34.8 Approximations for Large Parameters
34.8.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),$ $j_{1},j_{2},j_{3}\gg l_{3}\gg 1$,
34.8.2 $\cos\theta=\frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_% {1}+1)j_{2}(j_{2}+1)}},$
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). …