Just as with the Gauss hypergeometric function, particular cases of the local Heun function can be Liouvillian (that is, ``elementary'') functions. One way to obtain these functions is by pull-back transformations of Gauss hypergeometric equations with Liouvillian solutions. This paper presents the Liouvillian solutions of Heun's equations that are pull-backs of the parametric hypergeometric equations with cyclic or dihedral monodromy groups.
"Degenerate and dihedral Heun functions with parameters." Hokkaido Math. J. 45 (1) 93 - 108, February 2016. https://doi.org/10.14492/hokmj/1470080750