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

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Pith's one-line read Hyperbolic arcsine kernels plus Fourier filters extract quartic binomial-harmonic sums that evaluate to combinations of π and log(1+√2).

desk verdict Solid, carefully proved filter-plus-Mellin pipeline for quartic central-binomial harmonic sums; modest significance but real method and clean identities. read the letter →

arxiv 2607.09904 v1 pith:4DY6IU2J submitted 2026-07-10 math.NT math.CA

K. Srinivasa Raghava This is my paper · ORCID
classification math.NTmath.CA MSC 33C2011M0605A1040A25
keywords centralbinomialcoefficientharmonicnumberinversesinehypergeometricseriesroot-of-unityfilterlog-sineintegralBellpolynomialquarticsums
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Powers of the inverse sine produce series whose coefficients mix central binomials with finite repeated harmonic sums over squares. This paper packages those coefficients into two hyperbolic arcsine kernels and treats the kernels themselves as generating functions. Finite Fourier projection is applied to the kernels before any endpoint specialization; the quadratic case isolates the even (or odd) indices and thereby converts ordinary central binomials into quartic ones. The resulting closed forms are explicit linear combinations of powers of π and of L = log(1+√2). The same filtered kernels admit accelerated interior evaluations, Mellin deformations that produce denominator powers and polylogarithms at (√2-1)^{2}, and a square-law convolution that relates the two families. A sympathetic reader cares because the method systematically manufactures whole families of previously scattered identities from one analytic object rather than by case-by-case summation.

What carries the argument

The paired kernels O(z,x) = sinh(√z arcsin x)/√z and E(z,x) = [cosh(2√z arcsin x)-1]/z, which package the odd-square and ordinary-square coefficient families so that finite Fourier projection and Mellin deformation can be performed at the kernel level.

What would settle it

Direct high-precision numerical comparison of the partial sums of the four series in Theorem 6.1 against the claimed right-hand sides involving π and L, for several small weights m, would immediately confirm or refute the identities.

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Extended reading notes

Core claim

Projecting the two hyperbolic arcsine kernels with the quadratic root-of-unity filter before specialization yields the four residue-class identities of Theorem 6.1; in particular the even-residue odd-kernel sum equals [π^{2m+1} + (-1)^m (2L)^{2m+1}] / [2^{2m+2}(2m+1)!] for every m ≥ 0, with L = arsinh 1.

Load-bearing premise

The principal-branch choices for logarithm, square root and inverse sine, together with the claim that the Abel radial limits of the generating functions at ±1 equal the closed forms used in the filter, must hold; if those continuous boundary values failed, the quartic evaluations would collapse.

Editorial extensions

If this is right

  • Every weight-m quartic central-binomial series of the filtered type evaluates in closed form to a combination of π^{2m} or π^{2m+1} and the corresponding power of L.
  • Mellin deformation of the same projected kernels produces companion identities whose denominators are powers of the linear terms and whose constants involve polylogarithms at (√2-1)^{2}.
  • Interior evaluation of the quadratic filter supplies accelerated series with explicit geometric tail bounds.
  • The square-law relation E = 2O^{2} converts any convolution identity between the two harmonic alphabets into an algebraic identity among binomial coefficients.
  • Root-of-unity filters of higher order (cubic, character-mod-4, …) generate analogous residue-class evaluations for other arithmetic progressions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same kernel-plus-filter pipeline should extend without essential change to other odd multiple-zeta alphabets once the corresponding hyperbolic generating functions are written down.
  • Direct comparison of the filtered _2F1-type closed forms with the _4F3 kernels of Appendix B may produce new transformation identities between the two hypergeometric families.
  • Because the Mellin parameter can be differentiated under the integral, the method automatically supplies generating functions for all higher log-sine moments of the filtered series.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper packages the classical odd-square and ordinary-square coefficient families arising in powers of arcsin into two hyperbolic kernels O(z,x) and E(z,x). Finite Fourier (root-of-unity and periodic-weight) projections and Mellin deformations are applied at the kernel level before endpoint specialization. The quadratic case extracts even/odd subsequences, converting central-binomial series into quartic identities involving binom(4r,2r), powers of π, and L = arsinh 1 = log(1+√2). The same projected kernels yield accelerated interior evaluations with explicit tails, denominator-power series, and logarithmic companions involving polylogarithms at (√2−1)². Supporting material includes the square-law relation E = 2O², branch/convergence lemmas, spectral truncations, and a comparison showing that direct replacement of binom(2n,n) by binom(4n,2n) produces a different ₄F₃ family.

Significance. If the derivations hold—as they appear to—the paper supplies a clean, reusable analytic framework that unifies residue-class, accelerated, and Mellin-deformed central-binomial harmonic identities under a single pair of generating kernels. The main quartic evaluations (Theorem 6.1) and their logarithmic companions (Theorem 9.1) are concrete and checkable; the supporting lemmas on principal branches, Abel limits, and termwise Mellin operations are written with care. The work sits squarely in the classical tradition of Lehmer, Borwein–Chamberland, and recent arcsine-series papers, while the pre-specialization filtering step is a genuine methodological contribution. Complete closed-form proofs (no numerical fitting, no free parameters) are a clear strength.

minor comments (6)
  1. A short paragraph in the introduction or conclusion comparing the filtered-kernel approach more explicitly with the arcsine-moment / Bell-polynomial methods of Dilcher–Vignat (refs. [6–8]) would help the reader locate the novelty.
  2. Notation for the finite repeated sums ζ_n({2}^m) versus the Riemann zeta function ζ(s) is declared early, but the two still collide visually in Sections 6 and 8; a typographic distinction (e.g., bold or a different letter for the finite sums) would reduce cognitive load.
  3. Section 9: the low-weight polylog evaluations (9.2)–(9.3) are carefully derived; adding one intermediate numerical check (or a one-line Magma/Mathematica verification note) for the Li₂(ρ²) and Li₃(ρ²) coefficients would aid independent confirmation.
  4. Theorem 7.2: the tail constants are explicitly labelled “admissible, not optimal.” A single sentence noting that sharper Stirling-type bounds are available would prevent readers from treating the constants as best possible.
  5. Appendix B: the direct-quartic ₄F₃ comparison is useful; a one-sentence remark that the filtered identities of Theorem 6.1 are not special cases of these ₄F₃ series would make the contrast even sharper.
  6. Minor typographical/spacing artefacts appear in the extracted text (author name, some binomial displays). These are presumably clean in the source but should be double-checked before final production.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: kernels equal hyperbolic closed forms by hypergeometric identities; filters are ordinary root-of-unity projectors applied before specialization.

full rationale

The derivation is self-contained and non-circular. The kernels O(z,x) and E(z,x) are defined by their series (2.8)–(2.9) and proved equal to the hyperbolic expressions (2.10)–(2.11) via the classical trigonometric evaluations of 2F1 in Lemma 2.2 and termwise differentiation/integration. Coefficient extraction then recovers the unfiltered arcsine-power identities (Corollary 2.4). Finite Fourier projection (Theorems 5.1–5.2) is the standard root-of-unity filter applied to the absolutely summable coefficient sequences of Lemma 4.2; the quadratic case q=2 simply isolates the even/odd subsequences, converting binom(2n,n) into binom(4r,2r) and yielding the target quartic sums of Theorem 6.1 after the principal-branch evaluations arcsin(1)=π/2 and arcsin(i)=iL of Lemma 4.3. Mellin deformations (Theorems 5.4, 8.1–8.3) and logarithmic companions (Theorem 9.1) are obtained by termwise integration/differentiation justified by the same normal-convergence and integrability lemmas (4.1, 4.4). No parameter is fitted to data and then re-presented as a prediction; no uniqueness theorem is imported from the author’s prior work; the square-law E=2O² (Theorem 3.1) is an elementary hyperbolic identity whose coefficient form is a consequence, not an assumption. The comparison with direct quartic 4F3 kernels (Appendix B) further shows that the filtered identities are independent of that alternative construction. The closed forms involving π, L and polylogs at ρ² therefore emerge as outputs of the projection, not as inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

Pure analytic number theory / special functions. No empirical free parameters. Background consists of standard complex analysis (principal branches, Abel limits, normal convergence) and classical hypergeometric evaluations of arcsin. The two kernels O(z,x) and E(z,x) are invented packaging devices, not new physical entities; they are defined by series and proved equal to elementary hyperbolic functions. All subsequent claims rest on these definitions plus the root-of-unity projector.

assumptions (4)
  • standard math Principal branch of Log, square root and arcsin with Arg in (−π,π] (eqs. 4.1–4.4)
    Used throughout Sections 4–10 to evaluate boundary values at roots of unity and to justify continuous radial limits.
  • standard math Gauss trigonometric identity and Euler transformation for _2F_1 (Lemma 2.2)
    Load-bearing for the closed forms of the kernels O and E in Theorem 2.3.
  • standard math Absolute convergence of the coefficient series on the unit circle (Lemma 4.2) permitting Abel limits and termwise root-of-unity projection
    Justifies interchange of sum and finite Fourier average in Theorems 5.1–5.2 and the main quartic identities.
  • standard math Local uniform convergence of Mellin integrals for Re s > −1 (odd) / > −2 (even) allowing differentiation under the integral (Lemma 4.4)
    Required for the denominator-power and logarithmic companions in Sections 8–9.
invented entities (1)
  • Hyperbolic arcsine kernels O(z,x) and E(z,x) independent evidence
    purpose: Package the odd-square and ordinary-square coefficient families so that finite Fourier projection and Mellin deformation can be performed before specialization.
    Defined by series (2.8)–(2.9) and proved equal to sinh(√z arcsin x)/√z and (cosh(2√z arcsin x)−1)/z. They are organizational devices, not new mathematical objects with independent existence claims.

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Cite this review

Pith. "Pith review of Hyperbolic Arcsine Kernels, Finite Fourier Filters, and Quartic Central Binomial Harmonic Sums." pith.science (2026). https://pith.science/paper/4DY6IU2J

@misc{pith2026260709904,
  author       = {Pith},
  title        = {Pith review of: Hyperbolic Arcsine Kernels, Finite Fourier Filters, and Quartic Central Binomial Harmonic Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DY6IU2J}},
  note         = {Machine review of arXiv:2607.09904}
}
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 \(\binom{4r}{2r}\), \(\pi\), 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.

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Reference graph

Works this paper leans on

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