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 $H_n$, $H_{2n}$, 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.

arXiv comment28 pages