Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions
Abstract
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.
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BibTeX
@misc{valangelis2026explicit,
title = {{Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions}},
author = {J. G. R. Valangelis and Ailton C. Nascimento and Helder A. S. Costa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38461},
eprint = {2609.38461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38461},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions
AU - J. G. R. Valangelis
AU - Ailton C. Nascimento
AU - Helder A. S. Costa
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38461
UR - https://arxiv.org/abs/2609.38461
AB - 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.
ER -