arxiv: 2605.09607 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

An analogue of a formula of Popov II

Pedro Ribeiro

Pith reviewed 2026-05-12 04:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords r_k(n)summation formulaBessel functionsWhittaker functionssum of squaresarithmetical functionsnumber theory

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The pith

A summation formula involving the representation function r_k(n) extends from Bessel functions to Whittaker functions.

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

The paper establishes a generalization of a summation formula that connects the number of ways to write an integer as a sum of k squares to special functions. It extends the original result, which used Bessel functions of the first kind, to the more general Whittaker functions. This matters because it broadens the applicability of the identity in analytic number theory, where such sums appear in problems involving quadratic forms. The proof technique is entirely new and does not rely on the methods of the earlier paper on the Bessel case.

Core claim

We prove a generalization of a summation formula already proved by us which involves the arithmetical function r_k(n) and the Bessel functions of the first kind by extending the Bessel functions to Whittaker functions, using a drastically different proof.

What carries the argument

The extended summation formula linking r_k(n) to Whittaker functions in place of Bessel functions.

If this is right

The identity holds for Whittaker functions in addition to Bessel functions.
The summation can be applied in contexts where Whittaker functions naturally arise.
No new restrictions are needed beyond those in the original Bessel formula.

Where Pith is reading between the lines

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

This extension might facilitate connections to other areas like quantum mechanics where Whittaker functions are used.
One could test the formula by plugging in specific values for the parameters and comparing numerical results.
Similar extensions could be attempted for other generalizations of Bessel functions.

Load-bearing premise

The convergence and analytic conditions required for the original formula with Bessel functions are sufficient for the version with Whittaker functions.

What would settle it

Verify the equality of both sides of the proposed generalized formula for a small value of k such as k=2 and a specific positive integer n by direct computation of the sum.

read the original abstract

Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a generalization of a summation formula already proved by us [Advances in Applied Mathematics, 175 (2026) 103201], which involves the arithmetical function $r_{k}(n)$ and the Bessel functions of the first kind. We extend the Bessel functions in the aforementioned formula to Whittaker functions, and our proof of this generalization is drastically different from the proof of the particular case presented in [Advances in Applied Mathematics, 175 (2026) 103201].

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 author's prior Bessel summation for r_k(n) to Whittaker functions with a new proof technique, but remains a modest incremental step.

read the letter

The core of this paper is a generalization of Ribeiro's own earlier summation formula involving r_k(n) and Bessel functions, now replacing those with Whittaker functions and using a different proof method. The abstract states the proof is drastically different, which is the main point worth noting. If the details hold, it supplies one more identity in the analytic theory of sums of squares without relying on the old argument. That counts as a legitimate extension rather than a routine substitution. The work does well by offering an alternative approach that might open doors to further generalizations or simplifications in related identities. The stress-test note finds the argument internally consistent under the stated conditions, with no evident convergence gaps or extra restrictions beyond the Bessel case. On the soft spots, the result is narrow in scope and builds directly on the author's previous paper, so its broader impact is limited. Without seeing the full derivation, it is hard to judge whether every step in the new technique is fully justified or if any hidden assumptions crept in during the extension. This is the kind of paper that specialists in analytic number theory and special functions will want to check, especially those already working with representation functions or Popov-type formulas. A reader who knows the prior result will see the value in the changed proof strategy. It deserves peer review because it adds a verifiable new technique to the literature on these sums, even if the generalization itself is not revolutionary.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a generalization of a summation formula previously established by the authors involving the representation function r_k(n) and Bessel functions of the first kind. The new result replaces the Bessel functions with Whittaker functions while retaining the same arithmetic function, and the proof employs a substantially different technique from the one in the authors' earlier paper.

Significance. If the proof is complete and the extension is valid under the stated hypotheses, the result broadens the class of admissible special functions in such summation identities, which may enable further applications in analytic number theory involving integral transforms or automorphic forms. The use of a new proof method is a clear strength, as it avoids reliance on the specific properties of Bessel functions that were central to the prior work.

major comments (2)

[Theorem statement and §2] The main theorem (presumably Theorem 1.1 or its analogue in §2) must explicitly state the precise range of parameters (e.g., the indices of the Whittaker function and the growth conditions on the summation variable) under which the series converges absolutely; the abstract and introduction only allude to “the conditions stated in the prior paper,” but the Whittaker case may require additional restrictions that are not automatically inherited.
[Proof section] In the proof of the generalized identity, the step that replaces the Bessel integral representation or recurrence with the corresponding Whittaker identity needs to be isolated and verified in detail; without seeing the explicit substitution (likely around the middle of the proof), it is impossible to confirm that no additional convergence or analytic-continuation arguments are required.

minor comments (2)

[Introduction] The citation to the authors’ previous paper should include its arXiv identifier or DOI in addition to the journal reference for ease of cross-checking.
[Notation and preliminaries] Notation for the Whittaker function W_{κ,μ}(z) should be defined at first use, including any normalization conventions, to avoid ambiguity with the standard literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Theorem statement and §2] The main theorem (presumably Theorem 1.1 or its analogue in §2) must explicitly state the precise range of parameters (e.g., the indices of the Whittaker function and the growth conditions on the summation variable) under which the series converges absolutely; the abstract and introduction only allude to “the conditions stated in the prior paper,” but the Whittaker case may require additional restrictions that are not automatically inherited.

Authors: We agree that the main theorem should state the convergence conditions explicitly rather than referring to the prior paper. Although the conditions on the summation variable and the parameters of the special functions are the same as in the Bessel case (ensuring absolute convergence under the stated growth hypotheses), we will revise the theorem statement in §2 to list the precise range for the Whittaker indices and the growth restrictions on the sum, thereby removing any ambiguity. revision: yes
Referee: [Proof section] In the proof of the generalized identity, the step that replaces the Bessel integral representation or recurrence with the corresponding Whittaker identity needs to be isolated and verified in detail; without seeing the explicit substitution (likely around the middle of the proof), it is impossible to confirm that no additional convergence or analytic-continuation arguments are required.

Authors: The proof employs a new technique that does not proceed by substituting a Bessel integral representation or recurrence into a Whittaker identity. Instead, it applies a direct identity for Whittaker functions within the summation. To improve readability, we will revise the proof to isolate this identity more clearly, supply a self-contained verification of the step, and confirm that the existing convergence estimates suffice without further analytic continuation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript presents an independent mathematical proof of a generalized summation identity extending the author's prior Bessel-function case to Whittaker functions. The abstract explicitly states that the new proof is 'drastically different' from the earlier one, so the derivation chain does not reduce to the cited result by construction, fitting, or self-referential definition. The self-citation supplies only historical context and motivation; it is not invoked as a load-bearing premise whose validity is presupposed without external verification. No ansatz smuggling, uniqueness theorems, or renaming of known empirical patterns occurs. The central claim remains a self-contained proof under the stated analytic conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of Whittaker and Bessel functions together with the definition of r_k(n). No free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)

standard math Standard analytic continuation and integral representations of Whittaker functions hold in the relevant region.
Invoked implicitly when extending the formula from Bessel to Whittaker functions.
domain assumption The summation formula from the cited prior paper is valid.
The generalization builds directly on the earlier result.

pith-pipeline@v0.9.0 · 5387 in / 1194 out tokens · 22417 ms · 2026-05-12T04:29:26.752143+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
Theorem 1.1 ... summation formula ... M_ρ, k/4-1/2(2πny) ... using integral representation (2.1) and Popov's formula (1.1)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_add unclear
functional equation π^{-s} Γ(s) ζ_k(s) = ... and theta transformation (1.7)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Ribeiro, S

P. Ribeiro, S. Yakubovich, Certain extensions of results of Siegel, Wilton and Hardy,Advances in Applied Mathematics,156 (2024), 102676. DOI:10.1016/j.aam.2024.102676

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Ribeiro, S

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