is taken so that the characteristic exponents for the regular singularity at infinity are α and β (see below).
Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β).
The q-analog of Heun's equation has been discovered by Hahn (1971) and studied by Takemura (2017).
Heun's equation has a group of symmetries of order 192, isomorphic to the Coxeter group of the Coxeter diagram D4, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer.
The symmetries fixing the local Heun function form a group of order 24 isomorphic to the symmetric group on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points.