Arcsine kernels, inverse-binomial harmonic sums, and classical hypergeometric transformations
Abstract
We study inverse-central-binomial series with finite reciprocal-square factors and harmonic weights involving Hn, H2n, and 1/n. An arcsine integral gives explicit odd-zeta evaluations and a classification of coefficientwise cancellation within a polynomial ansatz. The scalar examples include an evaluation of weight seven and a direct proof of Sun's Conjecture 3.10. An entire generating relation organizes a second family with finite rational coefficient formulas. We then derive that relation, a supplementary generating family, and the complete two-order construction from Bailey's cubic transformation, contiguity, Gauss's quadratic and connection formulas, Euler's transformation, and Clausen's identity. The derivation retains all three infinity branches and resolves the vanishing denominators in the lower contiguous value. A doubling relation with a unique analytic solution and an explicit algebraic identity complete the proof. Thus the evaluations have a proved classical origin; no independent new hypergeometric summation principle is claimed. An application of a known complete-monotonicity property gives strict signs and global Taylor bounds. All endpoint limits, parameter derivatives, and coefficient extractions are justified.
Related
- Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
- Multiple harmonic star sums involving central binomial coefficients as weights
- Harmonic Catalan Number Series via Half-Integer Binomial Coefficients
- Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions
- Hypergeometric Solutions for Variants of the q-Hypergeometric Equation
BibTeX
@misc{raghava2026arcsine,
title = {{Arcsine kernels, inverse-binomial harmonic sums, and classical hypergeometric transformations}},
author = {K. Srinivasa Raghava},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.09904},
eprint = {2607.09904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09904},
}
RIS
TY - UNPB
TI - Arcsine kernels, inverse-binomial harmonic sums, and classical hypergeometric transformations
AU - K. Srinivasa Raghava
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2607.09904
UR - https://arxiv.org/abs/2607.09904
AB - We study inverse-central-binomial series with finite reciprocal-square factors and harmonic weights involving H_n, H_2n, and 1/n. An arcsine integral gives explicit odd-zeta evaluations and a classification of coefficientwise cancellation within a polynomial ansatz. The scalar examples include an evaluation of weight seven and a direct proof of Sun's Conjecture 3.10. An entire generating relation organizes a second family with finite rational coefficient formulas. We then derive that relation, a supplementary generating family, and the complete two-order construction from Bailey's cubic transformation, contiguity, Gauss's quadratic and connection formulas, Euler's transformation, and Clausen's identity. The derivation retains all three infinity branches and resolves the vanishing denominators in the lower contiguous value. A doubling relation with a unique analytic solution and an explicit algebraic identity complete the proof. Thus the evaluations have a proved classical origin; no independent new hypergeometric summation principle is claimed. An application of a known complete-monotonicity property gives strict signs and global Taylor bounds. All endpoint limits, parameter derivatives, and coefficient extractions are justified.
ER -