# §31.4 Solutions Analytic at Two Singularities: Heun Functions

For an infinite set of discrete values $q_{m}$, $m=0,1,2,\dots$, of the accessory parameter $q$, the function $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$, and hence also throughout the disk $|z|. To emphasize this property this set of functions is denoted by

 31.4.1 $\mathop{(0,1)\mathit{Hf}_{m}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,% \delta;z\right),$ $m=0,1,2,\dots$.

The eigenvalues $q_{m}$ satisfy the continued-fraction equation

 31.4.2 $q=\cfrac{a\gamma P_{1}}{Q_{1}+q-\cfrac{R_{1}P_{2}}{Q_{2}+q-\cfrac{R_{2}P_{3}}{% Q_{3}+q-\cdots}}},$

in which $P_{j},Q_{j},R_{j}$ are as in §31.3(i).

More generally,

 31.4.3 $\mathop{(s_{1},s_{2})\mathit{Hf}_{m}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,% \gamma,\delta;z\right),$ $m=0,1,2,\dots$,

with $(s_{1},s_{2})\in\{0,1,a,\infty\}$, denotes a set of solutions of (31.2.1), each of which is analytic at $s_{1}$ and $s_{2}$. The set $q_{m}$ depends on the choice of $s_{1}$ and $s_{2}$.

The solutions (31.4.3) are called the Heun functions. See Ronveaux (1995, pp. 39–41).