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arxiv: 2604.05895 · v1 · submitted 2026-04-07 · 🧮 math.NT · math.CO

Asymptotic expansions of integrals and Nielsen's polylogarithms

Markus Kuba , Moti Levy This is my paper

Pith reviewed 2026-05-10 18:15 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords asymptotic expansionsNielsen polylogarithmsmultiple zeta valueszeta valuesbinomial transformAppell sequencesintegral asymptoticsrandom variables
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The pith

Integrals of the form ∫ f(u) (1 + q u^n)^{w/n} du from 0 to 1 have complete asymptotic expansions as n approaches infinity, with coefficients given by Nielsen's generalized polylogarithms.

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

This paper derives full asymptotic expansions for integrals ∫ from 0 to 1 f(u) (1 + q u^n)^{w/n} du as n tends to infinity, for real w not zero and q in (-1,1], or positive w for q=-1. The coefficients in these expansions are expressed using Nielsen's generalized polylogarithms. When q equals -1, the expansion involves multiple zeta values that reduce to ordinary zeta values. For q=1, the coefficients involve alternating multiple zeta values, but under a symmetry constraint on the coefficient sequence, they reduce to polynomials in ordinary zeta values. The authors show this symmetry holds for several classical families such as Euler, Bernoulli, Genocchi, and Hermite numbers by translating it to a binomial transform property. They also obtain results on the convergence of norms of associated random variables.

Core claim

The integrals admit full asymptotic expansions whose coefficients are Nielsen polylogarithms. For q=-1 these reduce to zeta values. For q=1 a symmetry condition on the coefficients ensures reduction of alternating MZVs to polynomials in zeta values, and this condition is verified for Appell-type families via their binomial transforms.

What carries the argument

The relation of asymptotic coefficients to Nielsen's generalized polylogarithms, together with the symmetry constraint on the coefficient sequence that is equivalent to a property of the binomial transform.

If this is right

  • The asymptotic expansions are explicit and complete for the given parameter ranges.
  • For q=-1 the coefficients are rational multiples of ordinary zeta values.
  • Under the symmetry constraint for q=1 the coefficients become polynomials in ordinary zeta values.
  • The symmetry condition holds for the Euler, Bernoulli, Genocchi, and Hermite families.
  • Convergence results follow for the norms of the random variables defined in terms of these integrals.

Where Pith is reading between the lines

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

  • Similar asymptotic techniques could apply to other families of integrals or generating functions in analytic number theory.
  • The reduction to zeta values suggests possible new relations among multiple zeta values when symmetry is present.
  • The connection to random variable norms may yield applications in probability theory for large deviation or concentration results.
  • Extending the symmetry verification to other Appell or Sheffer sequences could produce more examples.

Load-bearing premise

The integrand function f must be such that the integral admits a full asymptotic expansion, and for the zeta polynomial reduction the coefficient sequence must satisfy the symmetry constraint.

What would settle it

Numerical computation of the integral for large n with a chosen f(u) and q=1 where the symmetry fails, showing that the coefficient does not match a polynomial in zeta values but requires an irreducible alternating MZV.

read the original abstract

This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the coefficients of the asymptotic expansions to Nielsen's generalized polylogarithms. For $q=-1$, we obtain an expansion in terms of multiple zeta values, which in this setting, reduce to ordinary zeta values. A key point is that for $q=1$, the integrals typically produce alternating multiple zeta values; we formulate a precise symmetry constraint on the relevant coefficient sequence under which all coefficients reduce to polynomials in ordinary zeta values. We also translate this symmetry into a statement about a binomial transform, and we verify the condition for several classical Appell-type families, like Euler, Bernoulli, Genocchi, and Hermite. Finally, we obtain precise results about the convergence of norms of random variables.

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

0 major / 2 minor

Summary. The paper derives full asymptotic expansions for integrals of the form ∫_0^1 f(u) (1 + q · u^n)^{w/n} du as n→∞, with real w≠0 and q∈(-1,1], or positive w for q=-1. Coefficients are related to Nielsen's generalized polylogarithms S_{n,p}(q). For q=-1 the expansion involves multiple zeta values reducing to ordinary zeta values. For q=1, under a symmetry constraint on the coefficient sequence (formulated via binomial transform), alternating MZVs reduce to polynomials in zeta values; this is verified explicitly for the generating functions of the Euler, Bernoulli, Genocchi, and Hermite families. The paper also obtains precise results on the convergence of norms of random variables.

Significance. If the derivations hold, the work supplies a systematic asymptotic framework for this family of integrals by expressing coefficients through Nielsen polylogarithms and providing explicit reductions to zeta values under verifiable symmetry conditions. The concrete checks on classical Appell-type sequences and the random-variable norm results give the claims external grounding and potential applicability in analytic number theory and probability. The approach relies on standard asymptotic expansions (substitution t = -n log u near u=1) and prior definitions rather than circular fitting.

minor comments (2)
  1. Abstract: the phrase 'precise results about the convergence of norms of random variables' is stated without any indication of the statement or theorem; a one-sentence summary of the main conclusion would improve readability.
  2. The symmetry constraint is introduced as a 'precise symmetry constraint on the relevant coefficient sequence'; an explicit equation (e.g., a_k = sum binom terms) in the statement of the main theorem would make the reduction to zeta polynomials immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our manuscript, as well as for the positive assessment of its significance and the recommendation of minor revision. The description correctly identifies the main results on asymptotic expansions, the role of Nielsen polylogarithms, the reductions to zeta values under symmetry conditions, and the applications to classical sequences and random-variable norms.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard methods and external verifications

full rationale

The paper derives the asymptotic expansions by substituting t = -n log u near u=1 and expressing the resulting moments through the independently defined generating function for Nielsen polylogarithms S_{n,p}(q). For q=-1 the series reduces to MZVs via known parity relations that collapse to single zeta values. For q=1 the binomial-transform symmetry is formulated as a condition and then explicitly verified on the external classical generating functions of the Euler, Bernoulli, Genocchi and Hermite families; these verifications rely on independent sequences rather than any fitted parameter or self-referential loop. All steps operate under the stated analyticity assumptions on f and the given ranges of w and q, with no interchange of limits or summation left unjustified. No load-bearing step reduces to a definition of its own output or to a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions and properties of Nielsen polylogarithms, multiple zeta values, and asymptotic expansion techniques from prior literature; no free parameters are fitted, no new entities postulated.

axioms (2)
  • standard math Standard analytic properties of integrals and asymptotic expansions as n→∞ hold under the given parameter restrictions on w and q.
    Invoked to justify existence of the full expansions and coefficient extractions.
  • standard math Nielsen polylogarithms and multiple zeta values satisfy their known functional equations and reduction relations.
    Used to express and simplify the expansion coefficients.

pith-pipeline@v0.9.0 · 5480 in / 1425 out tokens · 44666 ms · 2026-05-10T18:15:51.406594+00:00 · methodology

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