The violation of a uniqueness theorem and an invariant in the application of Poincar\'{e}--Perron theorem to Heun's equation

Abstract

The domain of convergence of a Heun function obtained through the Poincar\'{e}--Perron (P--P) theorem is not absolute convergence but conditional one [2]. We show that a uniqueness theorem is not available if we apply the P--P theorem into the Heun's equation. We verify that the uniqueness theorem is only applicable when a local Heun function is absolutely convergent.Comment: 10 pages, 5 figures. Change the title and abstract. The number of pages has been reduced. Correct some of the problems with Englis