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##### 2: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_{j}$, $j=1,2,\dots,N$, and at $\infty$, is given by
31.14.1 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.
The exponents at the finite singularities $a_{j}$ are $\{0,{1-\gamma_{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$, where …With $a_{1}=0$ and $a_{2}=1$ the total number of free parameters is $3N-3$. …
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
##### 3: DLMF Project News
error generating summary
##### 4: 26.5 Lattice Paths: Catalan Numbers
26.5.1 $C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}% \genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}% {0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n% -1}{n+1}.$
26.5.3 $C\left(n+1\right)=\sum_{k=0}^{n}C\left(k\right)C\left(n-k\right),$
26.5.4 $C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),$
26.5.5 $C\left(n+1\right)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\genfrac{(}{)}{0.0% pt}{}{n}{2k}2^{n-2k}C\left(k\right).$
26.5.7 $\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$
##### 5: 31.15 Stieltjes Polynomials
If $t_{k}$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}$ are the zeros of $S^{\prime}(z)$ (the derivative of $S(z)$), then $t_{k}$ is either a zero of $S^{\prime}(z)$ or a solution of the equation … The zeros $z_{k}$, $k=1,2,\ldots,n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\ifrac{\partial G}{\partial\zeta_{k}=0}$, $k=1,2,\ldots,n$, where … then there are exactly $\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}$ polynomials $S(z)$, each of which corresponds to each of the $\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}$ ways of distributing its $n$ zeros among $N-1$ intervals $(a_{j},a_{j+1})$, $j=1,2,\dots,N-1$. … If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$, where each $m_{j}$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_{j}$ zeros in the open interval $(a_{j},a_{j+1})$ for each $j=1,2,\dots,N-1$. … Let $S_{\mathbf{m}}(z)$ and $S_{\mathbf{l}}(z)$ be Stieltjes polynomials corresponding to two distinct multi-indices $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$ and $\mathbf{l}=(\ell_{1},\ell_{2},\dots,\ell_{N-1})$. …
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##### 7: 26.8 Set Partitions: Stirling Numbers
$s\left(n,k\right)$ denotes the Stirling number of the first kind: $(-1)^{n-k}$ times the number of permutations of $\{1,2,\ldots,n\}$ with exactly $k$ cycles. … where ${\left(x\right)_{n}}$ is the Pochhammer symbol: $x(x+1)\cdots(x+n-1)$. … For $n\geq 1$, … uniformly for $n=o\left(k^{1/2}\right)$. For asymptotic approximations for $s\left(n+1,k+1\right)$ and $S\left(n,k\right)$ that apply uniformly for $1\leq k\leq n$ as $n\to\infty$ see Temme (1993) and Temme (2015, Chapter 34). …
##### 8: 24.5 Recurrence Relations
24.5.1 $\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},$ $n=2,3,\dots$,
24.5.6 $\sum_{k=2}^{n}{n\choose k-2}\frac{B_{k}}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$ $n=2,3,\dots$,
24.5.7 $\sum_{k=0}^{n}{n\choose k}\frac{B_{k}}{n+2-k}=\frac{B_{n+1}}{n+1},$ $n=1,2,\dots$,
24.5.8 $\sum_{k=0}^{n}\frac{2^{2k}B_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$ $n=1,2,\dots$.
##### 9: 3.6 Linear Difference Equations
Miller (Bickley et al. (1952, pp. xvi–xvii)) that arbitrary “trial values” can be assigned to $w_{N}$ and $w_{N+1}$, for example, $1$ and $0$. …
###### Example 1. Bessel Functions
The Weber function $\mathbf{E}_{n}\left(1\right)$ satisfies … The values of $w_{n}$ for $n=1,2,\dots,10$ are the wanted values of $\mathbf{E}_{n}\left(1\right)$. … For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
##### 10: 26.6 Other Lattice Path Numbers
$D(m,n)$ 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 $(1,1)$. … $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …
26.6.10 $D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$ $m,n\geq 1$,