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The domain of convergence of a Heun function obtained through thePoincar\'{e}--Perron (P--P) theorem is not absolute convergence but conditionalone [2]. We show that a uniqueness theorem is not available if we apply theP--P theorem into the Heun's equation. We verify that the uniqueness theorem isonly applicable when a local Heun function is absolutely convergent.

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