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arxiv: 2402.07740 · v3 · pith:N6KCJ5ORnew · submitted 2024-02-12 · 🧮 math.HO

The double gamma function and Vladimir Alekseevsky

Yury A. Neretin This is my paper

Pith reviewed 2026-05-25 08:45 UTC · model grok-4.3

classification 🧮 math.HO
keywords double gamma functionVladimir AlekseevskyWeierstrass productJacobi theta functionspecial functionshistory of mathematicsq-gamma functionmultiple gamma functions
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The pith

Vladimir Alekseevsky discovered the double gamma function in 1888-89 by taking a quarter of the factors from the Weierstrass product of the Jacobi theta function.

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

This paper establishes that the double gamma function, now important across mathematics and physics for its many identities parallel to the gamma function, was originally discovered and investigated by Vladimir Alekseevsky in 1888-89. It traces the construction back to the Weierstrass product for sin(πx), where half the factors yield 1/Γ(x), half the factors of the Jacobi theta function θ₁ yield the q-gamma function, and the positive-quadrant quarter of those factors yields the double gamma. The paper also covers the brief continuation of this work until 1907 and Alekseevsky's biography. A sympathetic reader would care because it restores the historical origin of a function that reappeared decades later and now appears for many reasons in current research.

Core claim

Considering the Weierstrass product for the doubly quasiperiodic Jacobi theta function and taking the quarter of the factors corresponding to the positive quadrant in the lattice of periods, Alekseevsky arrived at his double gamma function. This parallels the earlier constructions of the reciprocal gamma function from the sine product and the q-gamma from half the theta factors, and the paper presents Alekseevsky's 1888-89 publications as the original detailed investigation of the double gamma.

What carries the argument

The quarter-selection of factors from the Weierstrass product for the Jacobi theta function θ₁, corresponding to the positive quadrant of the period lattice n₁ω + n₂ω₂.

If this is right

  • The double gamma function satisfies a large and amusing collection of identities parallel to those of the classical gamma function.
  • The double gamma appears for different reasons in many branches of mathematics and in mathematical physics.
  • Work on the double gamma continued with publications by Ernest Barnes, Jean Beaupin, Godfrey Hardy, and Vladimir Steklov in 1899-1907.
  • Publications on the topic stopped after 1907, and the double gamma reappeared due to Takuro Shintani roughly 70 years later.

Where Pith is reading between the lines

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

  • If the account is accurate, some of the identities satisfied by the double gamma may be traceable to Alekseevsky's original factor-selection arguments rather than later rediscoveries.
  • The quadrant-selection method from Weierstrass products of quasiperiodic functions could be examined for defining other multiple gamma functions beyond the double case.

Load-bearing premise

The paper's reading of Alekseevsky's 1888-89 publications as containing the original discovery of the double gamma via this specific quarter-factor construction depends on the accuracy of the historical sources it cites.

What would settle it

A primary-source analysis showing that Alekseevsky's publications did not describe the double gamma through quarter selection from the theta Weierstrass product, or documentation of an earlier independent construction of the same function.

read the original abstract

This paper is about a forgotten function and a forgotten mathematician. The double gamma function is now an important special function, which appears for different reasons in many branches of mathematics and in mathematical physics, as it satisfies a large and amusing collection of identities parallel to the classical gamma function. It was discovered and investigated in detail by Vladimir Alekseevsky in 1888-89. We outline his starting point here: considering the Weierstrass product for the entire periodic function $\sin \pi x$ and taking half of the factors corresponding to non-positive roots of the sine, we obtain the function $1/\Gamma(x)$. Considering the Weierstrass product for the doubly quasiperiodic Jacobi theta function $\vartheta_1$ and taking the half of factors we come to the $q$-gamma function (which appears, with a slight change in notation, in Eduard Heine's work). Considering the quarter of the factors (corresponding to the positive quadrant in the lattice $n_1\omega+n_2\omega_2$ of periods), Alekseevsky arrived at his double gamma function. The work in this direction was continued by Ernest Barnes, Jean Beaupin, Godfrey Hardy, and Vladimir Steklov in 1899--1907. After 1907, publications on this topic stopped, and the double gamma appeared again 70 years later due to Takuro Shintani. In this paper we discuss Alekseevsky's seminal work and its genesis, the history of the double gamma, and Alekseevsky's biography (1858-1916).

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

2 major / 1 minor

Summary. The paper claims that Vladimir Alekseevsky discovered and investigated the double gamma function in 1888-89 by starting from the Weierstrass product for sin(πx) to obtain 1/Γ(x), then from the product for the Jacobi theta function ϑ₁ to obtain the q-gamma function (with a slight change in notation from Heine), and finally by taking the quarter of the factors corresponding to the positive quadrant in the period lattice to arrive at the double gamma; it traces the subsequent development by Barnes, Beaupin, Hardy, and Steklov until 1907, the 70-year hiatus until Shintani, and includes Alekseevsky's biography (1858-1916).

Significance. If the historical attribution holds, the manuscript recovers an overlooked origin story for the double gamma function that parallels the classical Weierstrass constructions of the gamma and q-gamma functions, thereby contributing to the historiography of multiple gamma functions and special functions more broadly; it also restores visibility to Alekseevsky's contributions in a field where the function later proved important in mathematics and mathematical physics.

major comments (2)
  1. [Abstract] Abstract: the central claim that Alekseevsky arrived at the double gamma by taking the quarter of the factors from the Weierstrass product for ϑ₁ (corresponding to the positive quadrant in the lattice n₁ω + n₂ω₂) is presented as the direct genesis of his 1888-89 work, yet the manuscript supplies no direct quotations, page references, or verbatim excerpts from Alekseevsky's publications to confirm that this specific lattice-quadrant truncation was his starting point.
  2. [The section outlining Alekseevsky's starting point] The section outlining Alekseevsky's starting point and the 1888-89 timeline: the interpretive reconstruction of the progression from the sine product to the theta product to the quarter-factor double gamma rests on unshown primary sources, so the fidelity of the narrative to the original texts cannot be assessed from the manuscript alone.
minor comments (1)
  1. [Abstract] Abstract: the phrasing 'taking the half of factors we come to the q-gamma function' is slightly ambiguous in proportion; a brief clarification of the exact selection rule (and its relation to the subsequent quarter for the double gamma) would improve precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below, with planned revisions to strengthen the presentation of primary sources.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that Alekseevsky arrived at the double gamma by taking the quarter of the factors from the Weierstrass product for ϑ₁ (corresponding to the positive quadrant in the lattice n₁ω + n₂ω₂) is presented as the direct genesis of his 1888-89 work, yet the manuscript supplies no direct quotations, page references, or verbatim excerpts from Alekseevsky's publications to confirm that this specific lattice-quadrant truncation was his starting point.

    Authors: We agree that the abstract would be strengthened by explicit citations. The narrative follows the mathematical progression across Alekseevsky's 1888–89 publications, which move from the Weierstrass sine product to the theta product and then to the quadrant truncation. In revision we will add precise references to those papers together with page numbers and, where the texts permit, short excerpts illustrating the factor-selection steps. revision: yes

  2. Referee: [The section outlining Alekseevsky's starting point] The section outlining Alekseevsky's starting point and the 1888-89 timeline: the interpretive reconstruction of the progression from the sine product to the theta product to the quarter-factor double gamma rests on unshown primary sources, so the fidelity of the narrative to the original texts cannot be assessed from the manuscript alone.

    Authors: The section presents a reconstruction based on the chronological order and content of Alekseevsky's publications. We accept that the primary sources must be cited explicitly for verifiability. We will revise the section to include full bibliographic details and page references to the 1888 and 1889 works, making the connection between the published constructions and the quadrant truncation transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: purely historical narrative without derivations or self-referential math claims

full rationale

The paper is a historical overview of Alekseevsky's 1888-89 work on the double gamma function, tracing its genesis from Weierstrass products for sine and theta functions. It presents no mathematical derivations, no equations that are claimed to predict or derive new results, no fitted parameters, and no load-bearing self-citations. The narrative relies on historical interpretation rather than any chain of 'predictions' or 'first-principles results' that could reduce to inputs by construction. No steps match the enumerated circularity patterns (self-definitional, fitted-input-called-prediction, etc.). This is a self-contained historical account with no internal reduction of claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a historical paper with no mathematical content, so it introduces no free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5812 in / 1066 out tokens · 43615 ms · 2026-05-25T08:45:54.024459+00:00 · methodology

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