发表机构
Universidade Federal do Piauí（皮奥伊联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过保留次主导离心项并构造Frobenius级数，解决了Schwarzschild径向方程超几何展开的五项递推障碍，实现了高精度显式解析表示。
AI 中文摘要
Schwarzschild外部时空中的大质量Klein-Gordon方程可约化为合流Heun型径向微分方程。我们通过严格保留一个次主导的离心项，给出了该方程的精确约化，该离心项保持了视界指数不变，但关键性地移动了附加参数。在Svartholm–Schmidt超几何展开框架内，我们推导出三项递推关系，并建立了修正的连分式相容性条件，将最小尾部与低端递推相匹配。至关重要的是，我们证明了物理紧致化坐标z=(r-1)/r将不规则奇点移至z=1，本质上使超几何展开转变为无法退化为三项的五项递推。为绕过这一障碍，我们通过在z=0处直接构造Frobenius级数来构建视界归一化的物理分支。该表示对精确径向ODE表现出高阶收敛性和接近精度下限的残差，同时阐明了为什么准正态模谱选择在不规则奇点端点处仍然是一个连接问题。
英文摘要
The massive Klein-Gordon equation on the Schwarzschild exterior reduces to a radial differential equation of confluent-Heun type. We present its exact reduction by rigorously retaining a subleading centrifugal term that preserves the horizon indicial exponents but critically shifts the accessory parameters. Within the Svartholm--Schmidt hypergeometric-expansion framework, we derive the three-term recurrence relation and establish the corrected continued-fraction compatibility condition matching the minimal tail to the lower-end recurrence. Crucially, we demonstrate that the physical compactification coordinate $z=(r-1)/r$ shifts the irregular singular point to $z=1$, inherently transforming the hypergeometric expansion into a five-term recurrence that cannot degenerate to three terms. To circumvent this obstruction, we construct the horizon-normalized physical branch via a direct Frobenius series at $z=0$. This representation exhibits high-order convergence and near-precision-floor residuals against the exact radial ODE, while clarifying why quasinormal-mode spectral selection remains a connection problem at the irregular singular endpoint.
Comments43 pages, 3 figures
