pith. sign in

arxiv: 2512.12535 · v2 · submitted 2025-12-14 · 🧮 math.NT

On p-adic incomplete Mellin transforms and p-adic incomplete gamma-functions

Paul Buckingham This is my paper

Pith reviewed 2026-05-16 22:48 UTC · model grok-4.3

classification 🧮 math.NT
keywords p-adic gamma functionsincomplete Mellin transformp-adic integral transformrecurrence relationsp-adic analysisnumber theoryspecial functions
0
0 comments X

The pith

A p-adic incomplete gamma function can be constructed for any nonzero rational r at all primes where |r|_p equals 1.

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

For a fixed nonzero rational r, earlier constructions of p-adic incomplete gamma functions required the strict condition that |r minus 1|_p is less than 1, which holds for only finitely many primes p. The present work relaxes the restriction to the weaker condition |r|_p equals 1, which fails for only finitely many primes. It achieves this by extending a previously introduced p-adic integral transform to two variables and treating the extension as a p-adic analogue of the incomplete Mellin transform. A p-adic integration-by-parts formula then establishes the necessary recurrence relations for the gamma functions, exactly as integration by parts does in the complex setting. The approach therefore makes the p-adic incomplete gamma functions available for almost all primes without additional convergence requirements.

Core claim

The paper constructs the p-adic incomplete gamma function Γ_p(s, r) whenever |r|_p equals 1 by means of a two-variable p-adic integral transform that serves as the p-adic analogue of the incomplete Mellin transform. This transform satisfies a p-adic integration-by-parts formula that directly produces the recurrence relations satisfied by the incomplete gamma functions, reproducing the structure of the complex theory in the p-adic setting.

What carries the argument

The two-variable p-adic integral transform, which extends the single-variable transform from the 2024 paper and functions as the p-adic incomplete Mellin transform to define the gamma functions through integration by parts.

Load-bearing premise

The two-variable extension of the p-adic integral transform satisfies a p-adic integration-by-parts formula that generates the recurrence relations for the incomplete gamma functions without convergence or analyticity issues.

What would settle it

An explicit computation, for a concrete prime p with |r|_p equals 1 and specific numerical values of s and r, that produces a function violating the expected recurrence relation would show that the construction does not work.

read the original abstract

Let $r$ be a non-zero rational number. In a paper in the Transactions of the AMS in 2023, O'Desky and Richman gave a construction of a $p$-adic incomplete gamma-function $\Gamma_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$. Aside from the special case where $r = 1$, only finitely many primes satisfy that condition for a given $r$, so it is desirable to lessen this restriction. In the present paper, we give a construction that works under the much weaker condition that $|r|_p = 1$ using a $p$-adic integral transform we introduced in our paper of 2024 in Acta Arithmetica, which we interpret here as a $p$-adic analogue of an incomplete Mellin transform. For any given $r$, the condition $|r|_p = 1$ holds for all \emph{except} finitely many primes $p$. Our approach emphasizes the parallels between the complex and $p$-adic constructions, explaining how a $p$-adic integration-by-parts formula takes the place of complex integration by parts in the proof of the recurrence relations for the $p$-adic incomplete gamma-functions. We introduce a two-variable $p$-adic transform for the task, extending our earlier $p$-adic integral transform.

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

3 major / 2 minor

Summary. The paper constructs p-adic incomplete Mellin transforms and associated incomplete gamma functions Γ_p(·,r) for nonzero rational r satisfying |r|_p=1 (covering all but finitely many primes p for fixed r). It does so by introducing a two-variable extension of the authors' 2024 Acta Arithmetica p-adic integral transform, interpreting the extension as a p-adic analogue of an incomplete Mellin transform, and deriving the expected recurrence relations for Γ_p via a p-adic integration-by-parts formula that replaces the classical complex integration by parts. This relaxes the stricter |r-1|_p<1 hypothesis from the 2023 O'Desky-Richman construction.

Significance. If the two-variable extension and the integration-by-parts formula are rigorously established, the result meaningfully enlarges the set of primes for which p-adic incomplete gamma functions are available, facilitating broader p-adic interpolation and comparison with complex analogues. The explicit parallel drawn between complex and p-adic constructions is a conceptual strength.

major comments (3)
  1. [§2 (two-variable transform)] The two-variable extension of the 2024 one-variable integral transform is asserted to exist as a p-adic measure for |r|_p=1, yet no explicit definition, measure-theoretic construction, or convergence argument for this case is supplied; the abstract and introduction merely state that the extension is introduced without deriving the requisite estimates.
  2. [§3 (integration by parts)] The p-adic integration-by-parts formula is claimed to yield the recurrence relations for Γ_p(·,r) without boundary terms when |r|_p=1. However, the derivation is not carried out explicitly for this regime (where |r-1|_p need not be small), so it is impossible to verify that no additional convergence restrictions arise or that the formula holds exactly as stated.
  3. [§4 (recurrence relations)] The recurrence relations themselves are stated but not verified against any concrete example or special case (e.g., r=1 or small p) that would confirm the construction reproduces known values or satisfies the functional equation under the weaker hypothesis.
minor comments (2)
  1. [Introduction] The citation to the 2023 Transactions of the AMS paper by O'Desky-Richman should include the precise title and page range for easy reference.
  2. [§2] Notation for the two-variable transform (e.g., the precise domain of the second variable) is introduced without a displayed equation; a numbered definition would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to improve clarity and explicitness where needed.

read point-by-point responses

  1. Referee: [§2 (two-variable transform)] The two-variable extension of the 2024 one-variable integral transform is asserted to exist as a p-adic measure for |r|_p=1, yet no explicit definition, measure-theoretic construction, or convergence argument for this case is supplied; the abstract and introduction merely state that the extension is introduced without deriving the requisite estimates.

    Authors: We agree that the two-variable extension requires a more explicit treatment. In the revised manuscript we will expand Section 2 to give the full definition of the two-variable p-adic integral transform, the associated measure on the appropriate product space, and the detailed p-adic estimates establishing its existence and continuity precisely when |r|_p=1. These estimates follow from the same techniques used in our 2024 Acta Arithmetica paper but are now written out for the two-variable case. revision: yes

  2. Referee: [§3 (integration by parts)] The p-adic integration-by-parts formula is claimed to yield the recurrence relations for Γ_p(·,r) without boundary terms when |r|_p=1. However, the derivation is not carried out explicitly for this regime (where |r-1|_p need not be small), so it is impossible to verify that no additional convergence restrictions arise or that the formula holds exactly as stated.

    Authors: We acknowledge that the integration-by-parts formula needs an explicit derivation under the weaker hypothesis |r|_p=1. In the revision we will insert a self-contained proof of the p-adic integration-by-parts formula in this regime, verifying that no boundary terms appear and that the same convergence holds without further restrictions on |r-1|_p. The argument uses the two-variable measure introduced in Section 2 and the p-adic continuity properties already established for |r|_p=1. revision: yes

  3. Referee: [§4 (recurrence relations)] The recurrence relations themselves are stated but not verified against any concrete example or special case (e.g., r=1 or small p) that would confirm the construction reproduces known values or satisfies the functional equation under the weaker hypothesis.

    Authors: The referee is correct that explicit checks against special cases are absent. While the general proof applies to these cases, we will add a short subsection (or appendix) containing verifications for r=1 (recovering the standard p-adic gamma function) and for small primes p with |r|_p=1, confirming that the recurrence relations hold and match known values. revision: yes

Circularity Check

1 steps flagged

Self-cited 2024 one-variable integral transform supplies core properties for the two-variable extension and recurrence derivations

specific steps
  1. self citation load bearing [Abstract]
    "using a p-adic integral transform we introduced in our paper of 2024 in Acta Arithmetica, which we interpret here as a p-adic analogue of an incomplete Mellin transform. [...] We introduce a two-variable p-adic transform for the task, extending our earlier p-adic integral transform."

    The recurrence relations for the p-adic incomplete gamma-functions are obtained via the p-adic integration-by-parts formula applied to the two-variable extension; the existence, convergence, and exact form of this formula (especially for |r|_p=1) are carried over from the 2024 self-citation rather than re-derived from external benchmarks or first principles within this manuscript.

full rationale

The paper's construction of Γ_p(·,r) for |r|_p=1 explicitly extends the author's 2024 Acta Arithmetica p-adic integral transform to two variables and invokes its integration-by-parts formula to obtain the recurrence relations. This makes the self-citation load-bearing for the foundational analytic properties, yet the present work adds an independent two-variable definition and interpretation as an incomplete Mellin transform, so the central claim retains non-tautological content. No self-definitional loops, fitted predictions, or uniqueness theorems imported from the same authors appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and properties of the p-adic integral transform introduced in the author's 2024 Acta Arithmetica paper; no new free parameters, invented entities, or ad-hoc axioms are visible in the abstract, but the p-adic integration-by-parts formula is taken as given from prior work.

axioms (1)
  • domain assumption The p-adic integral transform from Buckingham 2024 admits a two-variable extension that behaves analogously to an incomplete Mellin transform and satisfies a p-adic integration-by-parts formula.
    Invoked to replace complex integration by parts and obtain recurrence relations for Γ_p(·,r).

pith-pipeline@v0.9.0 · 5544 in / 1574 out tokens · 35091 ms · 2026-05-16T22:48:09.194444+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

  1. [1]

    William W. Adams. Transcendental numbers in theP-adic domain.Amer. J. Math., 88:279–308, 1966

    work page 1966

  2. [2]

    Interpolationp-adique.Bull

    Yvette Amice. Interpolationp-adique.Bull. Soc. Math. France, 92:117–180, 1964

    work page 1964

  3. [3]

    Newman.Complex analysis

    Joseph Bak and Donald J. Newman.Complex analysis. Undergraduate Texts in Mathematics. Springer, New York, third edition, 2010

    work page 2010

  4. [4]

    Factorial-type recurrence relations andp-adic incom- plete gamma functions.Acta Arith., 212(1):1–29, 2024

    Paul Buckingham. Factorial-type recurrence relations andp-adic incom- plete gamma functions.Acta Arith., 212(1):1–29, 2024

    work page 2024

  5. [5]

    J. W. S. Cassels. Global fields. InAlgebraic Number Theory (Proc. In- structional Conf., Brighton, 1965), pages 42–84. Academic Press, London, 1967

    work page 1965

  6. [6]

    Raabe’s formula forp-adic gamma and zeta functions.Ann

    Henri Cohen and Eduardo Friedman. Raabe’s formula forp-adic gamma and zeta functions.Ann. Inst. Fourier (Grenoble), 58(1):363–376, 2008

    work page 2008

  7. [7]

    Fonctions d’une variablep-adique.Ast´ erisque, (330):13–59, 2010

    Pierre Colmez. Fonctions d’une variablep-adique.Ast´ erisque, (330):13–59, 2010

    work page 2010

  8. [8]

    Xiaojian Li, Jay Reiter, Shiang Tang, Napoleon Wang, and Jin Yi.p-Adic incomplete gamma functions and Artin-Hasse-type series.P-Adic. Num. Ultrametr. Anal. Appl., 14:335–343, 2022. 40 Paul Buckingham

    work page 2022

  9. [9]

    Ram Murty and Purusottam Rath.Transcendental numbers

    M. Ram Murty and Purusottam Rath.Transcendental numbers. Springer, New York, 2014

    work page 2014

  10. [10]

    Springer-Verlag, Berlin, 1999

    J¨ urgen Neukirch.Algebraic number theory, volume 322 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder

    work page 1999

  11. [11]

    Derangements and thep-adic incomplete gamma function.Trans

    Andrew O’Desky and David Harry Richman. Derangements and thep-adic incomplete gamma function.Trans. Amer. Math. Soc., 376:1065–1087, 2023

    work page 2023

  12. [12]

    Robert.A Course inp-adic Analysis

    A.M. Robert.A Course inp-adic Analysis. Springer, 2000

    work page 2000

  13. [13]

    W. H. Schikhof.Ultrametric calculus, volume 4 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006. An introduction top-adic analysis, Reprint of the 1984 original [MR0791759]

    work page 2006

  14. [14]

    Washington.Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics

    Lawrence C. Washington.Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edi- tion, 1997. 41

    work page 1997