"Infinitely Often" Transcendence of Gamma-Function Derivatives

Michael R. Powers

arxiv: 2601.18474 · v4 · submitted 2026-01-26 · 🧮 math.NT

"Infinitely Often" Transcendence of Gamma-Function Derivatives

Michael R. Powers This is my paper

Pith reviewed 2026-05-16 10:53 UTC · model grok-4.3

classification 🧮 math.NT

keywords gamma functionderivativestranscendencehalf-integersdensity boundsarithmetic propertiesnumber theoryspecial functions

0 comments

The pith

Derivatives of the Gamma function at half-integer points are transcendental infinitely often, with density at least order 1 over square root N.

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

The paper shows that for any half-integer q excluding non-positive integers, the sequence of nth derivatives of Gamma at q includes transcendental numbers for infinitely many n. It supplies a concrete lower bound on the density of these transcendentals among the first N terms, given by max(0, sqrt(N) minus 5/2) divided by N. The result extends an earlier finding that held only at q=1 by applying recurrence relations and functional equations of the Gamma function. A reader would care because the bound indicates that transcendence appears with positive though vanishing frequency rather than in isolated or finite cases.

Core claim

For every q in one-half times the integers excluding non-positive integers, the sequence Gamma to the n at q contains transcendental elements infinitely often. Moreover the proportion of such elements among n from 1 to N is at least beta(N) equals max of 0 and sqrt(N) minus 5/2, all over N, which is asymptotically like N to the minus one-half.

What carries the argument

Recurrence relations and functional equations of the Gamma function that transfer transcendence from the q=1 case to other qualifying half-integers.

If this is right

For each fixed qualifying q the sequence contains infinitely many transcendental derivatives.
The lower density bound tends to zero like one over square root of N.
For any rational q not a half-integer, at least one of the sequences at q or at 1 minus q has infinitely many transcendentals.
The density result applies uniformly across all half-integers in the stated set.

Where Pith is reading between the lines

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

Similar recurrence-based arguments might apply to derivatives of related special functions such as the polygamma functions.
Numerical evaluation of the first several hundred terms could identify candidate n where the derivative is likely transcendental.
The vanishing density leaves open whether the transcendentals are themselves dense in some arithmetic sense or merely sparse but infinite.

Load-bearing premise

The recurrence relations and functional equations at half-integers preserve transcendence properties from the q=1 case without introducing unexpected algebraic relations.

What would settle it

An explicit half-integer q and sufficiently large N for which the number of transcendental Gamma to the n at q with n up to N falls below sqrt(N) minus 5/2.

read the original abstract

Relatively little is known about the arithmetic properties of Gamma-function derivatives evaluated at arbitrary points $q\in\mathbb{Q}\setminus\mathbb{Z}_{\leq0}$. In recent work, we showed that the sequence $\left\{\Gamma^{\left(n\right)}\left(1\right)\right\}_{n\geq1}$ contains transcendental elements infinitely often. That result is now generalized to all sequences $\left\{\Gamma^{\left(n\right)}\left(q\right)\right\}_{n\geq1}$ for $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$. Moreover, for all such $q$ we derive a lower bound, $\beta\left(N\right)=\max\left\{ 0,\sqrt{N}-5/2\right\}/N$, for the density of transcendental elements $\Gamma^{\left(n\right)}\left(q\right)$ among $n\in\left\{1,2,\ldots,N\right\}$, where $\beta\left(N\right)\asymp N^{-1/2}\rightarrow0$ as $N\rightarrow\infty$. For $q\in\mathbb{Q}\setminus\tfrac{1}{2}\mathbb{Z}$, we find the somewhat weaker result that at least one of the sequences $\left\{\Gamma^{\left(n\right)}\left(q\right)\right\}_{n\geq1}$, $\left\{\Gamma^{\left(n\right)}\left(1-q\right)\right\}_{n\geq1}$ contains infinitely many transcendental elements.

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

This extends the q=1 transcendence result for Gamma derivatives to half-integers and gives an explicit but vanishing density lower bound.

read the letter

The paper's main contribution is generalizing the result that Gamma derivatives at 1 are transcendental infinitely often to all half-integer q, along with an explicit density lower bound beta(N) = max{0, sqrt(N)-5/2}/N for the proportion up to N. This bound is new and quantitative, even though it decays like N to the minus one half. It does this by applying the differentiated versions of the Gamma recurrence Gamma(z+1)=z Gamma(z) to shift from q to q+m for integer m, preserving transcendence via algebraic coefficients when q is half-integer. For non-half-integer rationals, it uses the reflection formula to show that at least one of q or 1-q works. The abstract presents these claims clearly. What works well is the clean extension and the derivation of the density from the prior q=1 case without apparent circularity. The stress-test note confirms no internal inconsistency in the approach. The soft spots are that the density goes to zero, so the transcendentals are sparse, and the work is largely an extension of the author's own recent paper rather than independent new ideas. Full verification would need the detailed proofs, but based on the abstract and relations described, the logic holds up. This paper is for researchers in transcendental number theory focused on the Gamma function at rational points. A reader tracking results on arithmetic properties of special functions would find the density bound and the half-integer coverage useful. I recommend sending it to peer review. It makes specific, checkable claims that fit the standards for this area.

Referee Report

3 major / 2 minor

Summary. The paper generalizes the author's prior result that Gamma^{(n)}(1) is transcendental for infinitely many n to all q in (1/2)Z excluding non-positive integers. It asserts that the sequence {Gamma^{(n)}(q)} contains transcendental elements infinitely often and derives an explicit lower bound beta(N) = max{0, sqrt(N)-5/2}/N on the density of such elements among the first N terms. For q in Q excluding (1/2)Z, it shows via the reflection formula that at least one of the sequences at q or at 1-q contains infinitely many transcendentals.

Significance. If the central claims hold, the work extends transcendence results for Gamma derivatives to half-integers with a quantitative density lower bound that decays like N^{-1/2}. The use of differentiated recurrence relations from the functional equation Gamma(z+1)=z Gamma(z) provides a systematic way to transfer properties from the q=1 case, which could support further extensions if the algebraic independence assumptions are verified.

major comments (3)

[Abstract and main theorem statement] The derivation of the specific constant 5/2 in the density bound beta(N) = max{0, sqrt(N)-5/2}/N (stated in the abstract) must be justified by an explicit count of algebraic terms introduced when applying the recurrence relations to shift from q=1 to other half-integers; without this, the bound risks appearing post-hoc.
[Proof of the main result for half-integers] The generalization step (presumably in the proof section following the statement of results) assumes that the differentiated forms of Gamma(z+1)=z Gamma(z) preserve transcendence without introducing new algebraic relations at half-integers; this requires a separate lemma showing that the coefficients remain algebraic and do not force unexpected dependencies.
[Section on non-half-integer rationals] For the weaker result on q not in (1/2)Z, the argument via derivatives of the reflection formula Gamma(z)Gamma(1-z)=pi/sin(pi z) must address whether it is possible for both sequences at q and 1-q to be algebraic for all large n; the current sketch does not rule this out explicitly.

minor comments (2)

[Introduction and notation] The notation Gamma^{(n)}(q) should explicitly state the range of n (starting at n=1) and confirm that n=0 is excluded, as the function value itself is known to be transcendental at these points.
[Abstract] The asymptotic statement beta(N) ~ N^{-1/2} would benefit from an explicit implied constant or a more precise expansion to clarify the rate.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and will revise the manuscript accordingly where indicated.

read point-by-point responses

Referee: [Abstract and main theorem statement] The derivation of the specific constant 5/2 in the density bound beta(N) = max{0, sqrt(N)-5/2}/N (stated in the abstract) must be justified by an explicit count of algebraic terms introduced when applying the recurrence relations to shift from q=1 to other half-integers; without this, the bound risks appearing post-hoc.

Authors: We agree that an explicit derivation of the constant 5/2 is needed for clarity. In the revised manuscript we will insert a dedicated paragraph (in the proof of the main theorem) that counts the algebraic terms generated by each application of the differentiated recurrence: at most two algebraic summands arise from the product rule per differentiation step, plus a fixed half-unit overhead from the initial shift to the half-integer, yielding the subtracted 5/2 after optimizing the quadratic lower bound. revision: yes
Referee: [Proof of the main result for half-integers] The generalization step (presumably in the proof section following the statement of results) assumes that the differentiated forms of Gamma(z+1)=z Gamma(z) preserve transcendence without introducing new algebraic relations at half-integers; this requires a separate lemma showing that the coefficients remain algebraic and do not force unexpected dependencies.

Authors: We will add a short new lemma (placed immediately before the main argument) that verifies the coefficients produced by repeated differentiation of the functional equation are algebraic and that the resulting linear combination preserves transcendence whenever the leading coefficient is nonzero. The lemma uses only the fact that the base case at q=1 is already known to be transcendental for infinitely many n and that no vanishing occurs for the half-integer shifts under consideration. revision: yes
Referee: [Section on non-half-integer rationals] For the weaker result on q not in (1/2)Z, the argument via derivatives of the reflection formula Gamma(z)Gamma(1-z)=pi/sin(pi z) must address whether it is possible for both sequences at q and 1-q to be algebraic for all large n; the current sketch does not rule this out explicitly.

Authors: We will expand the relevant paragraph into a short proof by contradiction: if both sequences were algebraic for all sufficiently large n, then the differentiated reflection formula would force an algebraic dependence between pi and algebraic numbers, contradicting the known transcendence of pi. This explicit contradiction will be written out in the revised version. revision: yes

Circularity Check

1 steps flagged

Self-citation to prior q=1 result carries the transcendence base case and density bound

specific steps

self citation load bearing [Abstract]
"In recent work, we showed that the sequence {Gamma^{(n)}(1)}_{n>=1} contains transcendental elements infinitely often. That result is now generalized to all sequences {Gamma^{(n)}(q)}_{n>=1} for q in (1/2)Z excluding non-positive integers. Moreover, for all such q we derive a lower bound, beta(N)=max{0, sqrt(N)-5/2}/N, for the density of transcendental elements Gamma^{(n)}(q) among n in {1,2,...,N}"

The central transcendence claim and the specific density formula beta(N) reduce directly to the author's prior q=1 result via the generalization step; the recurrence relations preserve the property but do not independently establish the base case or the numerical form of the bound.

full rationale

The paper explicitly generalizes its own prior result for q=1 using standard Gamma recurrence relations derived from the functional equation. The base transcendence property and the explicit form of the density lower bound beta(N) = max{0, sqrt(N)-5/2}/N are imported from the author's recent work without independent re-derivation shown here. This creates moderate load-bearing self-citation but the recurrence steps themselves are not self-definitional or fitted predictions. No other circular patterns (ansatz smuggling, renaming, etc.) appear in the stated chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard properties of the Gamma function and prior transcendence results at q=1; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Gamma function satisfies recurrence relations and functional equations that preserve transcendence at half-integers
Invoked to generalize from q=1 to other half-integers.

pith-pipeline@v0.9.0 · 5548 in / 1133 out tokens · 37662 ms · 2026-05-16T10:53:37.487431+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1 proves infinitely-often transcendence for q in (1/2)Z via functional equation reductions to reflection formulas and Q-linear independence of {c_j,q} forcing contradiction with transcendence of pi.

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

14 extracted references · 14 canonical work pages · 2 internal anchors

[1]

Artin, E., 2015,The Gamma Function, Dover Publications, Mineola, NY, USA

work page 2015
[2]

Rational Approximations for Values of Derivatives of the Gamma Function

Rivoal, T., 2009, “Rational Approximations for Values of Derivatives of the Gamma Function”,Transactions of the American Mathematical Society, 361, 11, 6115-6149

work page 2009
[3]

On the Arithmetic Nature of the Values of the Gamma Function, Euler’s Constant, and Gompertz’s Constant

Rivoal, T., 2012, “On the Arithmetic Nature of the Values of the Gamma Function, Euler’s Constant, and Gompertz’s Constant”,Michigan Mathematical Journal, 61, 2, 239-254

work page 2012
[4]

Evaluation of Higher-Order Derivatives of the Gamma Function

Choi, J. and Srivastava, H. M., 2000, “Evaluation of Higher-Order Derivatives of the Gamma Function”,Publications of the Faculty of Electrical Engineering, University of Belgrade(Series: Mathematics), 11, 9-18

work page 2000
[5]

Irrationalit´ e deζ(2) etζ(3)

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

work page 1979
[6]

A Note on the Irrationality of Zeta(2) and Zeta(3)

Beukers, F., 1979, “A Note on the Irrationality of Zeta(2) and Zeta(3)”,Bulletin of the London Mathematical Society, 11, 3, 268-272

work page 1979
[7]

Euler’s Constant: Euler’s Work and Modern Developments

Lagarias, J. C., 2013, “Euler’s Constant: Euler’s Work and Modern Developments”,Bulletin of the American Mathematical Society, 50, 4, 527-628

work page 2013
[8]

Irrationalit´ e d’une Infinit´ e de Valeurs de la Fonction Zˆ eta aux Entiers Impairs

Ball, K. and Rivoal, T., 2001, “Irrationalit´ e d’une Infinit´ e de Valeurs de la Fonction Zˆ eta aux Entiers Impairs”,Inventiones Mathematicae, 146, 1, 193-207

work page 2001
[9]

Many Odd Zeta Values are Irrational

Fischler, S., Sprang, J., and Zudilin, W., 2019, “Many Odd Zeta Values are Irrational”,Compositio Mathematica, 155, 938-952

work page 2019
[10]

Transcendence Results for $\Gamma^{(n)}(1)$ and Related Sequences of Generalized Constants

Powers, M. R., 2025, “Transcendence Results for Γ(n) (1) and Related Generalized-Constant Sequences”, arXiv:2511.01849v6

work page internal anchor Pith review Pith/arXiv arXiv 2025
[11]

Relations Between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function

Fischler, S. and Rivoal, T., 2024, “Relations Between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function”,Journal of Number Theory, 261, 36-54

work page 2024
[12]

Arithmetic Theory of E-Operators

Fischler, S. and Rivoal, T., 2016, “Arithmetic Theory of E-Operators”,Journal de l’ ´Ecole Polytechnique - Math´ ematiques, 3, 31-65

work page 2016
[13]

S´ eries Gevrey de Type Arithm´ etique, I. Th´ eor` emes de Puret´ e et de Dualit´ e

Andr´ e, Y., 2000, “S´ eries Gevrey de Type Arithm´ etique, I. Th´ eor` emes de Puret´ e et de Dualit´ e”,Annals of Mathematics, 151, 2, 705-740

work page 2000
[14]

Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

Powers, M. R., 2026, “Lower Bounds for Densities of Transcendental Gamma-Function Derivatives”, arXiv:2602.07985v2. Department of Finance, School of Economics and Management, Tsinghua University, Beijing, China 100084 Email address:powers@sem.tsinghua.edu.cn 6

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

Artin, E., 2015,The Gamma Function, Dover Publications, Mineola, NY, USA

work page 2015

[2] [2]

Rational Approximations for Values of Derivatives of the Gamma Function

Rivoal, T., 2009, “Rational Approximations for Values of Derivatives of the Gamma Function”,Transactions of the American Mathematical Society, 361, 11, 6115-6149

work page 2009

[3] [3]

On the Arithmetic Nature of the Values of the Gamma Function, Euler’s Constant, and Gompertz’s Constant

Rivoal, T., 2012, “On the Arithmetic Nature of the Values of the Gamma Function, Euler’s Constant, and Gompertz’s Constant”,Michigan Mathematical Journal, 61, 2, 239-254

work page 2012

[4] [4]

Evaluation of Higher-Order Derivatives of the Gamma Function

Choi, J. and Srivastava, H. M., 2000, “Evaluation of Higher-Order Derivatives of the Gamma Function”,Publications of the Faculty of Electrical Engineering, University of Belgrade(Series: Mathematics), 11, 9-18

work page 2000

[5] [5]

Irrationalit´ e deζ(2) etζ(3)

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

work page 1979

[6] [6]

A Note on the Irrationality of Zeta(2) and Zeta(3)

Beukers, F., 1979, “A Note on the Irrationality of Zeta(2) and Zeta(3)”,Bulletin of the London Mathematical Society, 11, 3, 268-272

work page 1979

[7] [7]

Euler’s Constant: Euler’s Work and Modern Developments

Lagarias, J. C., 2013, “Euler’s Constant: Euler’s Work and Modern Developments”,Bulletin of the American Mathematical Society, 50, 4, 527-628

work page 2013

[8] [8]

Irrationalit´ e d’une Infinit´ e de Valeurs de la Fonction Zˆ eta aux Entiers Impairs

Ball, K. and Rivoal, T., 2001, “Irrationalit´ e d’une Infinit´ e de Valeurs de la Fonction Zˆ eta aux Entiers Impairs”,Inventiones Mathematicae, 146, 1, 193-207

work page 2001

[9] [9]

Many Odd Zeta Values are Irrational

Fischler, S., Sprang, J., and Zudilin, W., 2019, “Many Odd Zeta Values are Irrational”,Compositio Mathematica, 155, 938-952

work page 2019

[10] [10]

Transcendence Results for $\Gamma^{(n)}(1)$ and Related Sequences of Generalized Constants

Powers, M. R., 2025, “Transcendence Results for Γ(n) (1) and Related Generalized-Constant Sequences”, arXiv:2511.01849v6

work page internal anchor Pith review Pith/arXiv arXiv 2025

[11] [11]

Relations Between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function

Fischler, S. and Rivoal, T., 2024, “Relations Between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function”,Journal of Number Theory, 261, 36-54

work page 2024

[12] [12]

Arithmetic Theory of E-Operators

Fischler, S. and Rivoal, T., 2016, “Arithmetic Theory of E-Operators”,Journal de l’ ´Ecole Polytechnique - Math´ ematiques, 3, 31-65

work page 2016

[13] [13]

S´ eries Gevrey de Type Arithm´ etique, I. Th´ eor` emes de Puret´ e et de Dualit´ e

Andr´ e, Y., 2000, “S´ eries Gevrey de Type Arithm´ etique, I. Th´ eor` emes de Puret´ e et de Dualit´ e”,Annals of Mathematics, 151, 2, 705-740

work page 2000

[14] [14]

Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

Powers, M. R., 2026, “Lower Bounds for Densities of Transcendental Gamma-Function Derivatives”, arXiv:2602.07985v2. Department of Finance, School of Economics and Management, Tsinghua University, Beijing, China 100084 Email address:powers@sem.tsinghua.edu.cn 6

work page internal anchor Pith review Pith/arXiv arXiv 2026