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

TL;DR

This paper demonstrates that the Poincaré--Perron theorem's application to Heun's equation does not guarantee a uniqueness theorem unless the local Heun function converges absolutely, highlighting limitations in the convergence analysis.

Contribution

The paper reveals that the Poincaré--Perron theorem's application to Heun's equation does not ensure uniqueness unless the convergence is absolute, clarifying a key limitation.

Findings

The domain of convergence for Heun functions via P--P theorem is conditional.

A uniqueness theorem holds only for absolutely convergent local Heun functions.

Conditional convergence affects the applicability of the uniqueness theorem.

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.