pith. sign in

arxiv: 2603.21079 · v2 · submitted 2026-03-22 · 🧮 math.NT

Integral Representations for Multiple Ap\'ery-Like Series

Jorge Antonio Gonz\'alez Layja This is my paper

Pith reviewed 2026-05-15 01:36 UTC · model grok-4.3

classification 🧮 math.NT
keywords Apéry-like seriesintegral representationsDirichlet eta functionpolylogarithmsFourier expansionsmultiple seriesLegendre chi function
0
0 comments X

The pith

Multiple Apéry-like series admit integral representations in terms of polylogarithms that yield new Dirichlet eta identities.

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

This paper establishes integral representations for six families of multiple Apéry-like series. The derivations rely on repeated integration by parts and Fourier expansions to convert the series into expressions involving polylogarithms, Legendre chi functions, and inverse tangent integrals. These formulas allow recovery of known special values in terms of Dirichlet eta, beta, and lambda functions. A new result expresses one family of the series as linear combinations of products of Dirichlet eta values. Readers interested in series evaluations would care because the integrals provide a systematic way to access closed forms for these hard-to-evaluate objects.

Core claim

Using repeated integration by parts and Fourier expansions, integral representations are derived for six families of multiple Apéry-like series. The formulas are expressed in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. Known evaluations are recovered as special cases in terms of Dirichlet eta, beta, and lambda functions. A new identity is obtained that expresses a family of such series as linear combinations of products of Dirichlet eta values.

What carries the argument

Repeated integration by parts followed by Fourier expansions of the integrands, producing expressions in polylogarithms, Legendre chi functions, and inverse tangent integrals.

If this is right

  • Known evaluations of Apéry-like series are recovered as special cases of the integral formulas.
  • One family of multiple Apéry-like series equals linear combinations of products of Dirichlet eta values.
  • The integral representations involve polylogarithms, Legendre chi functions, and inverse tangent integrals.
  • The method applies uniformly across the six distinct families considered.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The integral representations could enable faster numerical evaluation of the series for large parameters by using quadrature on the closed-form integrals.
  • Similar integration techniques might apply to other families of multiple series not covered in this work.
  • Specializing the parameters in the new identity may produce relations among multiple zeta values.
  • Independent verification of the eta identity could use generating functions or other analytic continuations.

Load-bearing premise

The repeated integration by parts and subsequent Fourier expansions remain valid for the multiple series without introducing convergence problems or requiring additional restrictions on the parameters.

What would settle it

Numerical evaluation of a concrete multiple Apéry-like series to high precision for fixed small integer parameters, compared against the value from the proposed integral representation; a mismatch beyond floating-point error would disprove the representation.

read the original abstract

We derive integral representations for six families of multiple Ap\'ery-like series using repeated integration by parts and Fourier expansions. The resulting formulas are expressed in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. As applications, we recover several known evaluations as special cases of our results, expressed in terms of Dirichlet eta, beta, and lambda functions. In addition, we obtain a new identity expressing a family of such series as linear combinations of products of Dirichlet eta values.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives integral representations for six families of multiple Apéry-like series via repeated integration by parts followed by Fourier expansions. The resulting expressions are given in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. Applications recover known special values in terms of Dirichlet eta, beta, and lambda functions, and a new identity is presented expressing one family as linear combinations of products of Dirichlet eta values.

Significance. If the derivations are valid, the work extends classical techniques for single Apéry-like series to the multiple case, supplying explicit integral forms that facilitate evaluation. The recovery of known results serves as a consistency check, while the new eta-product identity constitutes a concrete addition to the literature on multiple zeta and eta values. The approach relies on standard analytic operations rather than ad-hoc parameters, which is a methodological strength.

major comments (1)
  1. [§3–4] The justification for interchanging the order of summation and integration in the Fourier-expansion step for the multiple series (likely §3 or §4) is not accompanied by explicit convergence estimates or parameter restrictions. For multiple sums the absolute convergence may fail in regions where the single-series case succeeds, and this needs to be addressed to support the claimed integral representations.
minor comments (2)
  1. [§2] Notation for the six families should be introduced with a single consolidated table or definition block early in the paper to improve readability.
  2. [§5] A brief remark on the range of parameters for which the Legendre chi and inverse-tangent-integral expressions remain valid would clarify the scope of the results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3–4] The justification for interchanging the order of summation and integration in the Fourier-expansion step for the multiple series (likely §3 or §4) is not accompanied by explicit convergence estimates or parameter restrictions. For multiple sums the absolute convergence may fail in regions where the single-series case succeeds, and this needs to be addressed to support the claimed integral representations.

    Authors: We agree that explicit justification is required for the multiple-series case. In the revised manuscript we will insert a new lemma (placed before the derivations in §3 and §4) that supplies uniform convergence estimates via the Weierstrass M-test on compact subsets of the relevant polydisks, together with the precise parameter restrictions (e.g., |z_i|<1 for the polylogarithm arguments) under which the interchange is valid. The single-series estimates already present in the literature are recovered as the special case of one variable, but the multiple case needs the additional domination argument we will now provide. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from the explicit series definitions of the multiple Apéry-like families and applies repeated integration by parts followed by Fourier expansions to obtain integral representations in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. These steps are standard classical operations performed directly on the input series; no parameters are fitted to data, no quantities are defined in terms of the outputs they are claimed to predict, and no load-bearing steps rely on self-citations or imported uniqueness theorems. The new eta-product identity is obtained as a special case of the derived representations, preserving an independent derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on the applicability of integration by parts and Fourier expansions to the multiple series; no free parameters, new entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • domain assumption Repeated integration by parts converts the multiple series into integrals without loss of validity.
    Invoked to obtain the integral representations from the series definitions.
  • domain assumption Fourier expansions of the resulting integrands yield the stated polylogarithm and chi-function expressions.
    Central step used to reach the closed forms in terms of polylogarithms, Legendre chi, and inverse tangent integrals.

pith-pipeline@v0.9.0 · 5364 in / 1311 out tokens · 49888 ms · 2026-05-15T01:36:19.070104+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    C. I. V˘ alean,(Almost) Impossible Integrals, Sums, and Series, Springer, Cham, 2019

    work page 2019

  2. [2]

    C. I. V˘ alean,More (Almost) Impossible Integrals, Sums, and Series, Springer, Cham, 2023

    work page 2023

  3. [3]

    Xu and J

    C. Xu and J. Zhao, Ap´ ery-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I,arXiv preprintarXiv:2202.06195v2, 2022

    work page arXiv 2022

  4. [4]

    Xu and J

    C. Xu and J. Zhao, Ap´ ery-type series and colored multiple zeta values,Advances in Applied Mathematics,153:102610, 2024

    work page 2024

  5. [5]

    Xu, On the proof of the Genˇ cev–Rucki conjecture for multiple Ap´ ery-like series,arXiv preprint arXiv:2510.09052, 2025

    C. Xu, On the proof of the Genˇ cev–Rucki conjecture for multiple Ap´ ery-like series,arXiv preprint arXiv:2510.09052, 2025

    work page arXiv 2025

  6. [6]

    D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient,The American Math- ematical Monthly,92(7):449–457, 1985

    work page 1985

  7. [7]

    I. J. Zucker, On the seriesP∞ k=1 2k k −1 k−n and related sums,Journal of Number Theory,20(1):92– 102, 1985

    work page 1985

  8. [8]

    I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, 8th edition, Academic Press, New York, 2014

    work page 2014

  9. [9]

    J. M. Borwein, D. J. Broadhurst and J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values, and Zeta Values,Experimental Mathematics,10(1):25–34, 2001

    work page 2001

  10. [10]

    J. A. G. Layja, Evaluating six Ap´ ery-like series of weight 5,Integral Transforms and Special Functions,37(4):249–265, 2026

    work page 2026

  11. [11]

    J. A. G. Layja, On the Evaluation of Ap´ ery-Like Series Involving Multiplet-Harmonic Star Sums, arXiv preprintarXiv:2601.20027, 2026

    work page arXiv 2026

  12. [12]

    Lewin,Polylogarithms and Associated Functions, North Holland, New York, 1981

    L. Lewin,Polylogarithms and Associated Functions, North Holland, New York, 1981

    work page 1981

  13. [13]

    M. E. Hoffman, Multiple harmonic series,Pacific Journal of Mathematics,152(2):275–290, 1992. 14

    work page 1992

  14. [14]

    M. E. Hoffman, An odd variant of multiple zeta values,Communications in Number Theory and Physics,13(3):529–567, 2019

    work page 2019

  15. [15]

    Cantarini and J

    M. Cantarini and J. D’Aurizio, On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums,Bollettino dell’Unione Matematica Italiana,12:623–656, 2019

    work page 2019

  16. [16]

    Genˇ cev and P

    M. Genˇ cev and P. Rucki, On a class of multiple Ap´ ery-like series and their reduction,Mediterranean Journal of Mathematics,22:195, 2025

    work page 2025

  17. [17]

    Ap´ ery, Irrationalit´ e deζ(2) etζ(3),Ast´ erisque,61:11–13, 1979

    R. Ap´ ery, Irrationalit´ e deζ(2) etζ(3),Ast´ erisque,61:11–13, 1979

    work page 1979

  18. [18]

    Chu, Alternating series of Ap´ ery-type for the Riemann zeta function,Contributions to Discrete Mathematics,15(3):108–116, 2020

    W. Chu, Alternating series of Ap´ ery-type for the Riemann zeta function,Contributions to Discrete Mathematics,15(3):108–116, 2020

    work page 2020

  19. [19]

    Chen and W

    X. Chen and W. Wang, Ap´ ery-type series via colored multiple zeta values and Fourier-Legendre series expansions,Journal of Symbolic Computation,134:102508, 2026. 15

    work page 2026