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arxiv: 2511.01849 · v7 · submitted 2025-11-03 · 🧮 math.NT · math.PR

Transcendence Results for Gamma^((n))(1) and Related Sequences of Generalized Constants

Michael R. Powers This is my paper

Pith reviewed 2026-05-18 00:57 UTC · model grok-4.3

classification 🧮 math.NT math.PR
keywords transcendencealgebraic independenceEuler-Mascheroni constantEuler-Gompertz constantGumbel distributionShidlovskii theoremgeneralized constants
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The pith

The η^(n) from Gumbel(0,1) moments are algebraically independent for all n ≥ 0.

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

This paper studies sequences of constants γ^(n), δ^(n), and η^(n) obtained from raw, conditional, and partial moments of the Gumbel(0,1) distribution. These sequences obey the generalized Hardy-type relation γ^(n) + δ^(n)/e = η^(n) for n ≥ 0, extending the classical connection between the Euler-Mascheroni constant and the Euler-Gompertz constant. Generating functions are used to show that γ^(2n) lies in the field Q[γ, γ^(2), …, γ^(2n−1)] for n ≥ 2 and that γ^(n) is transcendental for infinitely many n. Application of Shidlovskii’s theorem to the associated E-functions then establishes algebraic independence of the entire sequence η^(n), which immediately yields concrete transcendence statements for the pairs and triples involving γ^(n) and δ^(n). Parallel results are derived for the non-alternating analogues.

Core claim

Via generating functions from Gumbel(0,1) moments, γ^(2n) belongs to the rational field extension generated by γ and the preceding odd-indexed terms for n ≥ 2. The η^(n) are shown to be algebraically independent using Shidlovskii’s theorem on E-functions, hence each is transcendental. This independence implies that for every n ≥ 1 at least one member of the pairs {γ^(n), δ^(n)/e} and {γ^(n), δ^(n)} is transcendental, and at least two members of the triple {γ^(n), δ^(n)/e, δ^(n)} are transcendental. Both δ^(n)/e and δ^(n) are transcendental for infinitely many n, each with lower asymptotic density at least 1/2. Parallel transcendence results hold for the tilde versions satisfying the non-altr

What carries the argument

Shidlovskii’s theorem applied to the E-functions constructed from the generating functions of the partial moments η^(n) of the Gumbel(0,1) distribution.

Load-bearing premise

The generating functions or E-functions arising from the Gumbel(0,1) moments must satisfy the hypotheses of Shidlovskii’s theorem.

What would settle it

Explicitly exhibiting a nonzero polynomial with rational coefficients that is satisfied by any finite collection of the numerical values η^(n) would disprove the claimed algebraic independence.

read the original abstract

Neither the Euler-Mascheroni constant, $\gamma=0.577215...$, nor the Euler-Gompertz constant, $\delta=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two constants are related through a well-known equation of Hardy, equivalent to $\gamma+\delta/e=\textrm{Ein}(1)$, which recently has been generalized to $\gamma^{(n)}+\delta^{(n)}/e=\eta^{(n)},\:n\geq0$ for sequences of constants $\gamma^{(n)}$, $\delta^{(n)}$, and $\eta^{(n)}$ (derived respectively from raw, conditional, and partial moments of the $\textrm{Gumbel}(0,1)$ probability distribution). Investigating $\gamma^{(n)}=(-1)^{n}\Gamma^{(n)}(1),\:n\geq1$ through $\textrm{Gumbel}(0,1)$ generating functions, we find that $\gamma^{(2n)}\in\mathbb{Q}[\gamma,\gamma^{(2)}$, $\gamma^{(3)},...,\gamma^{(2n-1)}]$ for $n\geq2$ and $\gamma^{(n)}$ is transcendental infinitely often. We then show, via a theorem of Shidlovskii, that the $\eta^{(n)}$ are algebraically independent, and therefore transcendental, for all $n\geq0$, implying that at least one element of each pair, $\left\{\gamma^{(n)},\delta^{(n)}/e\right\}$ and $\left\{\gamma^{(n)},\delta^{(n)}\right\}$, and at least two elements of the triple $\left\{\gamma^{(n)},\delta^{(n)}/e,\delta^{(n)}\right\}$ are transcendental for all $n\geq1$. Further analysis of the $\gamma^{(n)}$ and $\eta^{(n)}$ reveals that both the $\delta^{(n)}/e$ and $\delta^{(n)}$ are transcendental infinitely often with lower asymptotic densities of at least 1/2. Finally, we provide parallel results for the sequences $\widetilde{\delta}^{(n)}$ and $\widetilde{\eta}^{(n)}$ satisfying the "non-alternating analogue" equation $\gamma^{(n)}+\widetilde{\delta}^{(n)}/e=\widetilde{\eta}^{(n)}$.

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 paper defines sequences γ^(n), δ^(n), and η^(n) from raw, conditional, and partial moments of the Gumbel(0,1) distribution, generalizing the Hardy relation γ + δ/e = Ein(1) to γ^(n) + δ^(n)/e = η^(n) for n ≥ 0. It derives polynomial relations showing γ^(2n) lies in Q[γ, γ^(2), …, γ^(2n-1)] for n ≥ 2 and that γ^(n) is transcendental for infinitely many n. The central result applies Shidlovskii's theorem to establish algebraic independence of all η^(n) (n ≥ 0), implying that at least one element of each pair {γ^(n), δ^(n)/e} and {γ^(n), δ^(n)} and at least two elements of the triple {γ^(n), δ^(n)/e, δ^(n)} are transcendental for n ≥ 1. Parallel results are given for the non-alternating analogues, along with lower asymptotic densities of at least 1/2 for the transcendence of δ^(n)/e and δ^(n).

Significance. If the application of Shidlovskii's theorem is fully justified, the algebraic independence of the η^(n) would constitute a substantial advance in transcendence theory for generalized Euler-Mascheroni and Gompertz constants, yielding the first results on algebraic independence for these sequences and strengthening the known fact that at least one of γ or δ is transcendental. The polynomial relations among the γ^(n) and the density statements for transcendence of the δ sequences are also of interest.

major comments (1)
  1. [Application of Shidlovskii's theorem to the η^(n)] The section (or paragraph) applying Shidlovskii's theorem to the η^(n) series does not explicitly derive the linear differential equation system satisfied by the associated generating functions or E-functions, nor does it verify that the coefficients lie in Q(z) (rather than a larger extension) and that the Siegel-Shidlovskii irreducibility condition holds. These verifications are load-bearing for the algebraic-independence claim and the consequent transcendence statements for the pairs and triples involving γ^(n) and δ^(n).
minor comments (2)
  1. The explicit integral or moment definitions of γ^(n), δ^(n), and η^(n) should be stated in the introduction for immediate clarity before the generating-function setup is introduced.
  2. Notation for the non-alternating sequences ~δ^(n) and ~η^(n) is introduced late; a brief forward reference in the abstract or introduction would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for more explicit details in the application of Shidlovskii's theorem. We address this comment below and will revise the paper to include the requested verifications.

read point-by-point responses

  1. Referee: The section (or paragraph) applying Shidlovskii's theorem to the η^(n) series does not explicitly derive the linear differential equation system satisfied by the associated generating functions or E-functions, nor does it verify that the coefficients lie in Q(z) (rather than a larger extension) and that the Siegel-Shidlovskii irreducibility condition holds. These verifications are load-bearing for the algebraic-independence claim and the consequent transcendence statements for the pairs and triples involving γ^(n) and δ^(n).

    Authors: The referee correctly identifies that the manuscript does not explicitly derive the linear differential equation system or verify the conditions for Shidlovskii's theorem in full detail. This is a valid point, as these steps are indeed crucial for the rigor of the algebraic independence result. In the revised manuscript, we will add an appendix or expanded section that: derives the system of linear differential equations satisfied by the generating functions for the η^(n) from the integral representations or recurrence relations coming from the Gumbel(0,1) distribution; confirms that the coefficients are in Q(z); and verifies the Siegel-Shidlovskii irreducibility condition. With these additions, the application of the theorem will be fully justified, supporting the algebraic independence of the η^(n) and the consequent transcendence results. We do not anticipate any changes to the main theorems themselves. revision: yes

Circularity Check

0 steps flagged

No circularity: external Shidlovskii theorem applied to internally constructed E-functions

full rationale

The paper constructs the sequences γ^(n), δ^(n), η^(n) from Gumbel(0,1) moment generating functions and derives the polynomial relations γ^(2n) ∈ Q[γ, γ^(2), …, γ^(2n-1)] directly from those generating-function identities. It then invokes the independent external theorem of Shidlovskii to obtain algebraic independence of the η^(n). Because Shidlovskii’s result is a pre-existing theorem whose hypotheses are checked against the paper’s explicit series (rather than assumed or fitted from the target transcendence statements), and no step equates a claimed output to its own input by definition or self-citation chain, the derivation chain contains no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper defines the sequences from standard objects (Gamma derivatives and Gumbel moments) and invokes one external transcendence theorem; no numerical fitting or new physical entities are introduced.

axioms (2)
  • domain assumption The relation γ^(n) + δ^(n)/e = η^(n) holds for the sequences derived from raw, conditional, and partial moments of the Gumbel(0,1) distribution
    Generalization of Hardy's equation stated in the abstract as the starting point for all subsequent claims
  • domain assumption Shidlovskii's theorem applies to the generating functions or E-functions associated with the η^(n)
    Invoked directly to conclude algebraic independence
invented entities (1)
  • Sequences γ^(n), δ^(n), η^(n) and their non-alternating analogues no independent evidence
    purpose: Generalized constants obtained from higher-order moments of the Gumbel distribution
    Defined in the paper to extend the classical γ-δ relation; no independent existence claim beyond the definitions

pith-pipeline@v0.9.0 · 5965 in / 1685 out tokens · 55697 ms · 2026-05-18T00:57:54.384664+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. "Infinitely Often" Transcendence of Gamma-Function Derivatives

    math.NT 2026-01 unverdicted novelty 7.0

    For q in half-integers excluding non-positive integers, Gamma^(n)(q) is transcendental for a positive proportion of n up to N, at least max(0, sqrt(N)-5/2)/N many.

Reference graph

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