Hyperbolic Arcsine Kernels, Finite Fourier Filters, and Quartic Central Binomial Harmonic Sums

Abstract

Classical expansions of powers of the inverse sine contain central-binomial coefficients and finite repeated harmonic sums. We place the odd-square and ordinary-square coefficient families into two hyperbolic arcsine kernels and use these kernels as generating functions on which finite Fourier projection and Mellin deformation can be carried out before specialization. The quadratic projection extracts quartic subsequences and gives identities involving \(\binom{4r}{2r}\), \(π\), and \(L=\log(1+\sqrt2)\). The same projection admits accelerated interior forms and, after Mellin deformation, denominator-power and logarithmic companions with polylogarithms at \((\sqrt2-1)^2\). The paper also records the square-law convolution between the two kernels, periodic-weight filters, finite spectral truncations at negative square parameters, and the analytic details needed for branch choices, boundary convergence, and termwise Mellin operations. A final comparison shows that direct quartic kernels lead to a different \({}_4F_3\) family.