= Heun equation
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The general Heun equation is the second-order <linear ordinary differential equation>
$$
w''+\left(\frac\gamma z+\frac\delta{z-1}+\frac\epsilon{z-a}\right)w'+\frac{\alpha\beta z-q}{z(z-1)(z-a)}w=0,
\qquad \epsilon=\alpha+\beta+1-\gamma-\delta,\quad a\notin\{0,1\}.
$$
It is a canonical Fuchsian equation with four singular locations $0,1,a,\infty$, generically all regular singular points. Special parameter choices can make some singularities apparent or removable. The local exponent pairs are $(0,1-\gamma)$, $(0,1-\delta)$, $(0,1-\epsilon)$ and $(\alpha,\beta)$, respectively.<NIST Digital Library of Mathematical Functions>[Section31.2(i), equation31.2.1 and its exponent/parameter discussion]{ref}
The parameter $a$ locates the fourth singularity, while $q$ is an accessory parameter not fixed by those local exponent pairs. For instance, setting $\alpha=q=0$ makes the constant function $w=1$ a solution, as direct substitution shows. Coalescing singularities gives confluent Heun equations with different singularity structures; these are not the same four-distinct-point equation. Heun functions supply named solutions for boundary-value problems beyond the three-singularity hypergeometric case.
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