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##### 1: 34.2 Definition: $\mathit{3j}$ Symbol
###### §34.2 Definition: $\mathit{3j}$ Symbol
The quantities $j_{1},j_{2},j_{3}$ in the $\mathit{3j}$ 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: 26.10 Integer Partitions: Other Restrictions
$p\left(\mathcal{D},n\right)$ denotes the number of partitions of $n$ into distinct parts. $p_{m}\left(\mathcal{D},n\right)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. …$p\left(\mathcal{D}^{\prime}3,n\right)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. … Note that $p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)$, with strict inequality for $n\geq 9$. It is known that for $k>3$, $p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)$, with strict inequality for $n$ sufficiently large, provided that $k=2^{m}-1,m=3,4,5$, or $k\geq 32$; see Yee (2004). …
##### 3: 26.6 Other Lattice Path Numbers
###### Delannoy Number $D(m,n)$
$D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …
26.6.4 $r(n)=D(n,n)-D(n+1,n-1),$ $n\geq 1$.
26.6.10 $D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$ $m,n\geq 1$,
##### 4: 28.25 Asymptotic Expansions for Large $\Re z$
28.25.1 ${\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},$
$D_{-1}^{\pm}=0,$
$D_{0}^{\pm}=1,$
The upper signs correspond to ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$ and the lower signs to ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)$. The expansion (28.25.1) is valid for ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$ when …
##### 5: 1.11 Zeros of Polynomials
Set $z=w-\tfrac{1}{3}a$ to reduce $f(z)=z^{3}+az^{2}+bz+c$ to $g(w)=w^{3}+pw+q$, with $p=(3b-a^{2})/3$, $q=(2a^{3}-9ab+27c)/27$. … $f(z)=z^{3}-6z^{2}+6z-2$, $g(w)=w^{3}-6w-6$, $A=3\sqrt{4}$, $B=3\sqrt{2}$. Roots of $f(z)=0$ are $2+\sqrt{4}+\sqrt{2}$, $2+\sqrt{4}\rho+\sqrt{2}\rho^{2}$, $2+\sqrt{4}\rho^{2}+\sqrt{2}\rho$. … Let … Then $f(z)$, with $a_{n}\not=0$, is stable iff $a_{0}\not=0$; $D_{2k}>0$, $k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$; $\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}$, $k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor$.
##### 6: 28.8 Asymptotic Expansions for Large $q$
28.8.1 $\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$
Also let $\xi=2\sqrt{h}\cos x$ and $D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)$18.3). …
28.8.4 $U_{m}(\xi)\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi% \right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}% \left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)% \dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}% \left(\xi\right)\right)+\cdots,$
28.8.5 $V_{m}(\xi)\sim\frac{1}{2^{4}h}\bigg{(}-D_{m+2}\left(\xi\right)-m(m-1)D_{m-2}% \left(\xi\right)\bigg{)}+\frac{1}{2^{10}h^{2}}\left(D_{m+6}\left(\xi\right)+(m% ^{2}-25m-36)D_{m+2}\left(\xi\right)-m(m-1)(m^{2}+27m-10)D_{m-2}\left(\xi\right% )-6!\genfrac{(}{)}{0.0pt}{}{m}{6}D_{m-6}\left(\xi\right)\right)+\cdots,$
$\sigma_{m}\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^{3}% +59s}{2^{11}h^{3}}+\cdots,$
##### 7: 4.43 Cubic Equations
$A=\left(-\tfrac{4}{3}p\right)^{1/2},$
4.43.2 $z^{3}+pz+q=0$
• (a)

$A\sin a$, $A\sin\left(a+\frac{2}{3}\pi\right)$, and $A\sin\left(a+\frac{4}{3}\pi\right)$, with $\sin\left(3a\right)=4q/A^{3}$, when $4p^{3}+27q^{2}\leq 0$.

• (b)

$A\cosh a$, $A\cosh\left(a+\frac{2}{3}\pi i\right)$, and $A\cosh\left(a+\frac{4}{3}\pi i\right)$, with $\cosh\left(3a\right)=-4q/A^{3}$, when $p<0$, $q<0$, and $4p^{3}+27q^{2}>0$.

• (c)

$B\sinh a$, $B\sinh\left(a+\frac{2}{3}\pi i\right)$, and $B\sinh\left(a+\frac{4}{3}\pi i\right)$, with $\sinh\left(3a\right)=-4q/B^{3}$, when $p>0$.

• ##### 8: 19.21 Connection Formulas
The complete case of $R_{J}$ can be expressed in terms of $R_{F}$ and $R_{D}$: … $R_{D}\left(x,y,z\right)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using …
19.21.8 $R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},$
19.21.9 $xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_% {F}\left(x,y,z\right).$
Because $R_{G}$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\leq 0$ if the variables are real, thereby avoiding cancellations when $R_{G}$ is calculated from $R_{F}$ and $R_{D}$ (see §19.36(i)). …
##### 9: 1.10 Functions of a Complex Variable
Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup\partial D$. … If $u(z)$ is harmonic in $D$, $z_{0}\in D$, and $u(z)\leq u(z_{0})$ for all $z\in D$, then $u(z)$ is constant in $D$. Moreover, if $D$ is bounded and $u(z)$ is continuous on $\overline{D}$ and harmonic in $D$, then $u(z)$ is maximum at some point on $\partial D$. … Let $F(z)$ be a multivalued function and $D$ be a domain. … Suppose $D$ is a domain, and …
##### 10: 12.19 Tables
• Kireyeva and Karpov (1961) includes $D_{p}\left(x(1+i)\right)$ for $\pm x=0(.1)5$, $p=0(.1)2$, and $\pm x=5(.01)10$, $p=0(.5)2$, 7D.

• Karpov and Čistova (1964) includes $D_{p}\left(x\right)$ for $p=-2(.1)0$, $\pm x=0(.01)5$; $p=-2(.05)0$, $\pm x=5(.01)10$, 6D.

• Karpov and Čistova (1968) includes $e^{-\frac{1}{4}x^{2}}D_{p}\left(-x\right)$ and $e^{-\frac{1}{4}x^{2}}D_{p}\left(ix\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$, 7D or 8S.

• Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U\left(n-\tfrac{1}{2},x\right)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.