# §31.6 Path-Multiplicative Solutions

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

 31.6.1 $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$ $m=0,1,2,\dots$,

with $(s_{1},s_{2})\in\{0,1,a\}$, but with another set of $\{q_{m}\}$. 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 ${\mathrm{e}^{2\nu\pi i}}$. These solutions are called path-multiplicative. See Schmidt (1979).