# §31.14(i) Definitions

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 ${\frac{{d}^{2}w}{{dz}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_{j}}{z-a_{j}}% \right)\frac{dw}{dz}+\left(\sum_{j=1}^{N}\frac{q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.

The exponents at the finite singularities $a_{j}$ are $\{0,{1-\gamma_{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$, where

 31.14.2 $\displaystyle\alpha+\beta+1$ $\displaystyle=\sum_{j=1}^{N}\gamma_{j},$ $\displaystyle\alpha\beta$ $\displaystyle=\sum_{j=1}^{N}a_{j}q_{j}.$

The three sets of parameters comprise the singularity parameters $a_{j}$, the exponent parameters $\alpha,\beta,\gamma_{j}$, and the $N-2$ free accessory parameters $q_{j}$. With $a_{1}=0$ and $a_{2}=1$ the total number of free parameters is $3N-3$. Heun’s equation (31.2.1) corresponds to $N=3$.

# Normal Form

 31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
 31.14.4 $\frac{{d}^{2}W}{{dz}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{\gamma}_{j}}{(z-a_{% j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,$ $\sum_{j=1}^{N}\tilde{q}_{j}=0$,
 31.14.5 $\displaystyle\tilde{q}_{j}$ $\displaystyle=\frac{1}{2}\sum_{\substack{k=1\\ k\neq j}}^{N}\frac{\gamma_{j}\gamma_{k}}{a_{j}-a_{k}}-q_{j},$ $\displaystyle\tilde{\gamma}_{j}$ $\displaystyle=\frac{\gamma_{j}}{2}\left(\frac{\gamma_{j}}{2}-1\right).$

# §31.14(ii) Kovacic’s Algorithm

An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). The algorithm returns a list of solutions if they exist.

For applications of Kovacic’s algorithm in spatio-temporal dynamics see Rod and Sleeman (1995).