Computing Gamma(p/q) with Beta function values

Jan L\"ugering

arxiv: 2605.22888 · v1 · pith:TGAVXEELnew · submitted 2026-05-21 · 🧮 math.HO · math.NT

Computing Gamma(p/q) with Beta function values

Jan L\"ugering This is my paper

Pith reviewed 2026-05-25 02:54 UTC · model grok-4.3

classification 🧮 math.HO math.NT

keywords Gamma functionBeta functionKontsevich-Zagier periodsrational argumentsspecial valuesperiodstranscendental numbers

0 comments

The pith

Gamma(p/q) can be computed explicitly from Beta function values at rational arguments, which qualify as Kontsevich-Zagier periods.

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

This expository paper demonstrates explicit formulas that express the Gamma function evaluated at rational points p/q through values of the Beta function at rational arguments. These Beta values are treated as Kontsevich-Zagier periods, which places the Gamma computations inside an established algebraic framework for periods. A reader would care because the approach turns abstract special-function evaluations into concrete reductions that inherit period properties and algebraic relations. The presentation focuses on making the connection direct and usable rather than deriving new results from scratch.

Core claim

In this expository article we show explicitly how to compute Gamma(p/q) in terms of Beta function values which in turn are Kontsevich-Zagier Periods.

What carries the argument

The explicit reduction of Gamma(p/q) to Beta function values B(r,s) at rational r and s, which carry the period property.

If this is right

Gamma values at rational points inherit algebraic relations from the period framework via the Beta intermediary.
Numerical or symbolic computation of Gamma(p/q) reduces to evaluating specific Beta integrals or products at rationals.
The method supplies an explicit bridge allowing Gamma values to be studied using known period identities and linear relations.

Where Pith is reading between the lines

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

The reduction might allow transfer of transcendence or algebraic independence results from periods back to Gamma values at rationals.
Similar Beta-mediated links could be checked for other special functions whose values appear in period tables.

Load-bearing premise

Beta function values at rational arguments are Kontsevich-Zagier periods.

What would settle it

A concrete rational pair where the Beta value fails to satisfy the defining properties of a Kontsevich-Zagier period while the corresponding Gamma(p/q) is computed from it.

read the original abstract

In this expository article we show explicitly how to compute Gamma(p/q) in terms of Beta function values which in turn are Kontsevich-Zagier Periods.

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.

Desk Editor's Note private letter to a colleague

Expository note that spells out the classical Gamma-Beta identity at rationals and its link to periods, with no new content.

read the letter

The paper is an expository note that recalls how Gamma(p/q) follows from Beta function values at rational arguments via the standard identity B(x,y) = Gamma(x)Gamma(y)/Gamma(x+y), with those Beta values identified as Kontsevich-Zagier periods. That is the entire claim. It does nothing beyond laying out this known chain in one place. For someone who needs a quick pointer to the relation for computation or reference, the write-up could be convenient. The link to periods is correctly taken from existing theory rather than derived here. The central relations are standard and hold up. The soft spot is that the work asserts no new result, supplies no original derivation, and adds no explicit formulas or error checks beyond what is already in the literature. The abstract itself labels the piece expository, so the value is purely in the presentation. This is for readers already working with periods or rational Gamma values who might want the connection spelled out plainly. It does not rise to the level that would justify sending it to referees, since there is no new mathematics or evidence to evaluate. I would not bring it to a reading group or cite it.

Referee Report

1 major / 0 minor

Summary. The manuscript is an expository article that claims to show explicitly how to compute Gamma(p/q) for rational p/q in terms of Beta function values at rational arguments, which are in turn identified as Kontsevich-Zagier periods via the classical relation B(x,y) = Gamma(x)Gamma(y)/Gamma(x+y).

Significance. The central relations invoked are standard and correctly identified as such, with no free parameters, ad-hoc axioms, or circular constructions. The paper asserts no new theorems and supplies no original derivations, so its value is purely pedagogical in linking Gamma values at rationals to the period framework. This connection is already known in the literature on periods, limiting the significance to exposition.

major comments (1)

Abstract: the claim to 'show explicitly' the computation of Gamma(p/q) is not supported by any visible derivations, explicit formulas, examples, or error analysis in the manuscript. This is load-bearing for the central claim of the paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We agree that the manuscript is purely expository and contains no new theorems or derivations. We address the single major comment below by revising the abstract to align with the actual content.

read point-by-point responses

Referee: Abstract: the claim to 'show explicitly' the computation of Gamma(p/q) is not supported by any visible derivations, explicit formulas, examples, or error analysis in the manuscript. This is load-bearing for the central claim of the paper.

Authors: We agree with this observation. The manuscript recalls the standard relation B(x,y) = Gamma(x)Gamma(y)/Gamma(x+y) and notes that Beta values at rational arguments are Kontsevich-Zagier periods, without supplying original derivations, explicit closed-form expressions beyond this identity, numerical examples, or error analysis. The wording 'show explicitly' in the abstract is therefore inaccurate. We will change the abstract to: 'In this expository article we discuss how Gamma(p/q) for rational p/q may be expressed using values of the Beta function at rational arguments, which are Kontsevich-Zagier periods.' The body text will be reviewed to ensure it does not overstate the scope. No error analysis will be added, as none is claimed. revision: yes

Circularity Check

0 steps flagged

No circularity; expository recall of classical identities with external period status taken as given.

full rationale

The manuscript is explicitly expository and asserts no new theorem. Its central step is the classical identity B(x,y) = Γ(x)Γ(y)/Γ(x+y) applied at rational arguments, followed by the known (non-trivial but external) fact that such B(r,s) are Kontsevich-Zagier periods. Neither step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; both are standard and correctly flagged as such. No load-bearing derivation is claimed to be original, and the paper does not invoke any uniqueness theorem or ansatz from prior work by the same author. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard mathematical facts whose proofs lie outside the paper.

axioms (2)

standard math B(x, y) = Gamma(x) Gamma(y) / Gamma(x + y) for Re(x) > 0, Re(y) > 0
Functional equation relating Beta and Gamma, invoked to solve for Gamma(p/q) from a Beta value.
domain assumption Beta function values at rational arguments are Kontsevich-Zagier periods
This is a known theorem in the theory of periods; the paper uses it to classify the resulting expressions.

pith-pipeline@v0.9.0 · 5528 in / 1398 out tokens · 25376 ms · 2026-05-25T02:54:01.911629+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show explicitly how to recursively compute Gamma(p/q) in terms of Beta function values which in turn are Kontsevich-Zagier periods.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The second equation follows from the Legendre Duplication formula via the Beta function.

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

4 extracted references · 4 canonical work pages

[1]

Kontsevich and D

M. Kontsevich and D. Zagier,Periods, in Mathematics Unlimited—2001 and Beyond, Springer, 2001, pp. 771–808

work page 2001
[2]

Wilms,On the Faltings Height of the Curvey 2 =x n−1, arXiv preprint arXiv:2601.15271, 2026

R. Wilms,On the Faltings Height of the Curvey 2 =x n−1, arXiv preprint arXiv:2601.15271, 2026

work page arXiv 2026
[3]

Waldschmidt,Diophantine Properties of the Periods of the Fermat Curve, in Number Theory Related to Fermat’s Last Theorem, ed

M. Waldschmidt,Diophantine Properties of the Periods of the Fermat Curve, in Number Theory Related to Fermat’s Last Theorem, ed. N. Koblitz, Birkhäuser, 1982

work page 1982
[4]

Waldschmidt,Transcendence of Periods: The State of the Art, Pure and Applied Mathematics Quarterly, 2(2) (2006), 435–463

M. Waldschmidt,Transcendence of Periods: The State of the Art, Pure and Applied Mathematics Quarterly, 2(2) (2006), 435–463. 6

work page 2006

[1] [1]

Kontsevich and D

M. Kontsevich and D. Zagier,Periods, in Mathematics Unlimited—2001 and Beyond, Springer, 2001, pp. 771–808

work page 2001

[2] [2]

Wilms,On the Faltings Height of the Curvey 2 =x n−1, arXiv preprint arXiv:2601.15271, 2026

R. Wilms,On the Faltings Height of the Curvey 2 =x n−1, arXiv preprint arXiv:2601.15271, 2026

work page arXiv 2026

[3] [3]

Waldschmidt,Diophantine Properties of the Periods of the Fermat Curve, in Number Theory Related to Fermat’s Last Theorem, ed

M. Waldschmidt,Diophantine Properties of the Periods of the Fermat Curve, in Number Theory Related to Fermat’s Last Theorem, ed. N. Koblitz, Birkhäuser, 1982

work page 1982

[4] [4]

Waldschmidt,Transcendence of Periods: The State of the Art, Pure and Applied Mathematics Quarterly, 2(2) (2006), 435–463

M. Waldschmidt,Transcendence of Periods: The State of the Art, Pure and Applied Mathematics Quarterly, 2(2) (2006), 435–463. 6

work page 2006