# Džrbasjan sum

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##### 1: 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. $p\left(\mathcal{D}k,n\right)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …If more than one restriction applies, then the restrictions are separated by commas, for example, $p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$. … Note that $p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)$, with strict inequality for $n\geq 9$. …
##### 2: 28.25 Asymptotic Expansions for Large $\Re z$
28.25.1 ${\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},$
$D_{-1}^{\pm}=0,$
$D_{0}^{\pm}=1,$
28.25.3 $(m+1)D^{\pm}_{m+1}+{\left((m+\tfrac{1}{2})^{2}\pm(m+\tfrac{1}{4})8\mathrm{i}h+% 2h^{2}-a\right)D^{\pm}_{m}}\pm(m-\tfrac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1% }^{\pm}=0,$ $m\geq 0$.
##### 3: 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 …
##### 4: 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 $R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},$
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)).
19.21.11 $6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),$
##### 5: 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 $(1,1)$.
26.6.1 $D(m,n)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{m+n-% k}{n}=\sum_{k=0}^{n}2^{k}\genfrac{(}{)}{0.0pt}{}{m}{k}\genfrac{(}{)}{0.0pt}{}{% n}{k}.$
26.6.6 $\sum_{n=0}^{\infty}D(n,n)x^{n}=\frac{1}{\sqrt{1-6x+x^{2}}},$
##### 6: 29.15 Fourier Series and Chebyshev Series
29.15.34 $[D_{1},D_{3},\dots,D_{2n+1}]^{\mathrm{T}},$
29.15.35 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+1}^{2}+{\tfrac{1}{2}k^{2}% \left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{n-1}D_{2p+1}D_{2p+3}\right)=1},$
29.15.40 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}% \sum_{p=1}^{n}D_{2p}D_{2p+2}=1,$
##### 7: 1.16 Distributions
We denote it by $\mathcal{D}(I)$. … for all $\phi\in\mathcal{D}(I)$. … For any locally integrable function $f$, its distributional derivative is $Df=\Lambda^{\prime}_{f}$. … The distributional derivative $D^{k}f$ of $f$ is defined by …
##### 8: 29.6 Fourier Series
29.6.42 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}D_{2p+1}^{2}+{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{\infty}D_{2p+1}D_{2p+3}\right)=1},$
29.6.43 $\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0,$
29.6.53 $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=1}^{\infty}D_{2p}\sin\left(2p\phi\right),$
29.6.57 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2% }\sum_{p=1}^{\infty}D_{2p}D_{2p+2}=1,$
29.6.58 $\sum_{p=0}^{\infty}(2p+2)D_{2p+2}>0,$
##### 9: 1.11 Zeros of Polynomials
$D=b^{2}-4ac.$
The sum and product of the roots are respectively $-b/a$ and $c/a$. … Let
$D_{1}=a_{1},$
Then $f(z)$, with $a_{n}\not=0$, is stable iff $a_{0}\not=0$; $D_{2k}>0$, $k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$; $\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}$, $k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor$.
##### 10: 10.20 Uniform Asymptotic Expansions for Large Order
In the following formulas for the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, $u_{k}$, $v_{k}$ are the constants defined in §9.7(i), and $U_{k}(p)$, $V_{k}(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … Note: Another way of arranging the above formulas for the coefficients $A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)$, and $D_{k}(\zeta)$ would be by analogy with (12.10.42) and (12.10.46). … Each of the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, $k=0,1,2,\dotsc$, is real and infinitely differentiable on the interval $-\infty<\zeta<\infty$. … For numerical tables of $\zeta=\zeta(z)$, $(4\zeta/(1-z^{2}))^{\frac{1}{4}}$ and $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$ see Olver (1962, pp. 28–42). … The equations of the curved boundaries $D_{1}E_{1}$ and $D_{2}E_{2}$ in the $\zeta$-plane are given parametrically by …