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

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

The radius of convergence of the Heun function

Heun functions generalize well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric--type functions. They are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solutions of the Schrodinger equation of quantum mechanics, and addition of three quantum spins. We consider the radius of convergence of the Heun function, and we show why the Poincare-Perron (P-P) theorem is not available for the absolute convergence since it is applied to the Heun's equation. Moreover, we construct the absolute convergence test in which is suitable for the three term recurrence relation in a power series.Comment: 14 pages. 3 figures. The number of pages has been reduced. Give the reader greater insight into what the key results are in the introduction. Correct some of the problems with Englis