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##### 1: 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 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
31.14.4 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,$ $\sum_{j=1}^{N}\tilde{q}_{j}=0$,
##### 2: Bibliography C
• J. R. Cash and R. V. M. Zahar (1994) A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.
• F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert $W$ function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.
• Cunningham Project (website)
• ##### 3: 31.15 Stieltjes Polynomials
31.15.2 $\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,$ $k=1,2,\dots,n$.
31.15.3 $\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.$
31.15.6 $a_{j} $j=1,2,\dots,N-1$,
31.15.7 $q_{j}=\gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}},$ $j=1,2,\dots,N$.
31.15.8 $S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),$ $z_{j}\in(a_{j},a_{j+1})$,
##### 5: 26.11 Integer Partitions: Compositions
For example, there are eight compositions of 4: $4,3+1,1+3,2+2,2+1+1,1+2+1,1+1+2$, and $1+1+1+1$. $c\left(n\right)$ denotes the number of compositions of $n$, and $c_{m}\left(n\right)$ is the number of compositions into exactly $m$ parts. $c\left(\in\!T,n\right)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\ldots\}$. … The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
##### 6: 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 $\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$
##### 7: 27.18 Methods of Computation: Primes
These algorithms are used for testing primality of Mersenne numbers, $2^{n}-1$, and Fermat numbers, $2^{2^{n}}+1$. …
##### 8: 24.15 Related Sequences of Numbers
The Fibonacci numbers are defined by $u_{0}=0$, $u_{1}=1$, and $u_{n+1}=u_{n}+u_{n-1}$, $n\geq 1$. …
##### 9: 3.6 Linear Difference Equations
For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
##### 10: 26.6 Other Lattice Path Numbers
26.6.2 $M(n)=\sum_{k=0}^{n}\frac{(-1)^{k}}{n+2-k}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac% {(}{)}{0.0pt}{}{2n+2-2k}{n+1-k}.$
26.6.4 $r(n)=D(n,n)-D(n+1,n-1),$ $n\geq 1$.
26.6.10 $D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$ $m,n\geq 1$,
26.6.11 $M(n)=M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k),$ $n\geq 2$.
26.6.13 $M(n)=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}C\left(n+1-k\right),$