§31.6 Path-Multiplicative Solutions

Keywords:: Heun’s equation
Referenced by:: §31.11(v), §31.8
Permalink:: http://dlmf.nist.gov/31.6

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

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

Symbols:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $s_{j}$ : parameters
Permalink:: http://dlmf.nist.gov/31.6.E1
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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 $e^{{2\nu\pi i}}$ . These solutions are called path-multiplicative. See Schmidt (1979).