31 Heun FunctionsProperties31.5 Solutions Analytic at Three Singularities: Heun Polynomials31.7 Relations to Other Functions

A further extension of the notation (31.4.1) and (31.4.3) is given by

31.6.1 | $$\left({s}_{1},{s}_{2}\right){\mathit{Hf}}_{m}^{\nu}\left(a,{q}_{m};\alpha ,\beta ,\gamma ,\delta ;z\right),$$ | ||

$m=0,1,2,\mathrm{\dots}$, | |||

with $\left({s}_{1},{s}_{2}\right)\in \left\{0,1,a\right\}$, but with another set of $\left\{{q}_{m}\right\}$. This
denotes a set of solutions of (31.2.1) with the property that if we
pass around a simple closed contour in the $z$-plane that encircles ${s}_{1}$ and
${s}_{2}$ once in the positive sense, but not the remaining finite singularity,
then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. These
solutions are called *path-multiplicative*. See Schmidt (1979).