Heun functions are defined as a natural generalization of the hypergeometric function, to be the solutions of the Fuchsian differential equation. The author presents some recent progress on Heun functions, gathering results from classical analysis up to elliptic functions. The author describes Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and gives the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.
For the entire collection see [Zbl 1117.39001].