arxiv: 2603.21072 · v3 · submitted 2026-03-22 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Analytic Study of p-Bessel Functions: Fractional Calculus, Integral Representations, and Complex Extensions

Masaya Kitajima

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:40 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-Bessel functionsErdélyi-Kober fractional derivativesintegral representationsPoisson integralsoscillatory kernelsp-circleslattice point problemsanisotropic oscillations

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The pith

p-Bessel functions admit a hierarchical structure via Erdélyi-Kober fractional derivatives, explicit integral representations, and complex extensions via Poisson formulas.

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

This paper develops the analytic theory of the p-Bessel functions that arise from Fourier analysis on planar domains bounded by p-circles. It organizes these functions into a hierarchy through repeated application of Erdélyi-Kober-type fractional derivatives. The work supplies real-analytic integral representations that expose axis-dependent behavior and extends the functions to the complex domain by Poisson-type integral formulas. These constructions convert the functions from formal Fourier kernels into objects equipped with concrete expressions suitable for asymptotic and analytic work. A reader would care because the results supply the necessary analytic tools for studying oscillatory integrals over anisotropic p-norm domains and for lattice-point counting in those domains.

Core claim

The p-Bessel functions J^[p]_(ω,ϕ) form a family that can be generated recursively from base cases by Erdélyi-Kober fractional derivatives, possess explicit real-analytic integral representations adapted to the p-circle geometry, and admit analytic continuation into the complex plane through Poisson-type integral formulas. These properties establish the functions as new oscillatory kernels distinct from classical Bessel functions and furnish the analytic infrastructure needed for applications to p-circle lattice-point problems.

What carries the argument

The p-Bessel function J^[p]_(ω,ϕ), defined originally by Fourier analysis on domains bounded by p-circles with 0 < p ≤ 2 and 2/p a natural number, and reorganized hierarchically by Erdélyi-Kober fractional derivatives together with Poisson integral extensions.

If this is right

The functions now possess real-analytic integral representations that support direct study of axis-dependent asymptotics.
Poisson-type formulas provide a concrete route for complex extension and continuation.
The constructions supply a rigorous framework for anisotropic oscillatory phenomena.
The analytic properties lay the foundation for applications to lattice-point discrepancies in p-circle domains.

Where Pith is reading between the lines

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

The same fractional-calculus hierarchy may be adaptable to other families of kernels defined by Fourier transforms on non-Euclidean boundaries.
The integral representations could be used to derive new addition theorems or recurrence relations that parallel those of classical Bessel functions.
Verification of the representations against direct numerical Fourier integrals on sample astroid-type domains would provide an immediate consistency check.

Load-bearing premise

The p-Bessel functions are taken as already defined by Fourier analysis on the indicated p-circle domains.

What would settle it

Numerical evaluation of the derived integral representations for a concrete p-value, frequency, and phase that fails to recover the original Fourier coefficients computed directly on the p-circle domain.

Figures

Figures reproduced from arXiv: 2603.21072 by Masaya Kitajima.

**Figure 1.** Figure 1: Examples of the p-circle and the approximation by unit squares. The structure of this definition consists of an outer series in the radial variable r, whose coefficients Φ [p] k,φ encode angular anisotropy through a finite combinatorial sum. Note that this condition on p guarantees uniform convergence of the series representation on compact subsets of R≥0. These p-Bessel functions extend the classical Bess… view at source ↗

read the original abstract

We present a systematic analytic study of the $p$-Bessel functions $\mathcal{J}_{\omega,\varphi}^{[p]}$, a novel class of generalized Bessel functions arising from Fourier analysis on planar domains bounded by $p$-circles, including astroid-type shapes with $0<p\le2$ satisfying $(2/p)\in\mathbb{N}$. While previous work established Hardy-type oscillatory identities for these domains, expressing lattice point discrepancies via $p$-Bessel functions, the present paper focuses on the intrinsic analytic properties of the functions themselves. In particular, we (i) construct a hierarchical structure of $\{\mathcal{J}_{\omega,\varphi}^{[p]}\}_{\omega\ge0}$ using Erd\'{e}lyi-Kober-type fractional derivatives, (ii) derive explicit real-analytic integral representations suitable for investigating axis-dependent asymptotic behavior, and (iii) extend the functions to the complex domain through Poisson-type integral formulas. These results establish $p$-Bessel functions as genuinely new oscillatory kernels, providing a rigorous framework for studying anisotropic oscillatory phenomena and laying the analytic foundation for applications in $p$-circle lattice point problems.

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

The paper adds a fractional hierarchy, axis-dependent integrals, and complex extensions to p-Bessel functions, but everything rests on an earlier Fourier definition that is not re-derived or verified here.

read the letter

The main takeaway is that the authors organize their p-Bessel functions into a hierarchy via Erdélyi-Kober fractional derivatives, supply explicit real-analytic integral representations that separate axis behavior, and give a Poisson-type formula for the complex extension. These steps are presented as new relative to classical Bessel theory and the authors' prior oscillatory identities for p-circle domains. If the derivations are complete, the integral representations could be useful for extracting asymptotics in lattice-point problems on those shapes. The paper does a reasonable job keeping the focus on intrinsic analytic properties rather than immediate applications, which keeps the claims scoped. The central limitation is the reliance on the prior definition of the functions through Fourier analysis on p-circle domains for 0 < p ≤ 2 with 2/p a natural number. The abstract treats those functions as already given and regular, so any gaps in convergence or uniqueness from the earlier work would propagate directly. No sample formulas, error bounds, or verification steps appear in the summary, which makes it impossible to confirm the representations hold without the full text. This is narrow material aimed at analysts or number theorists already working on generalized oscillatory kernels or anisotropic lattice discrepancies. A specialist in that corner might borrow the integral forms for concrete calculations, but the paper will not interest a broader audience. I would send it to referees because the constructions are specific enough to be checked and the scope is modest, even though the foundational dependence needs explicit attention in review.

Referee Report

3 major / 2 minor

Summary. The paper claims to develop the intrinsic analytic properties of the p-Bessel functions J^{[p]}_{ω,φ} (defined via Fourier analysis on p-circle domains with 0 < p ≤ 2 and 2/p natural) by (i) building a hierarchy via Erdélyi-Kober fractional derivatives, (ii) obtaining explicit real-analytic integral representations for axis-dependent asymptotics, and (iii) extending to the complex plane via Poisson-type integrals, thereby positioning these functions as new oscillatory kernels for anisotropic phenomena and p-circle lattice-point applications.

Significance. If the claimed constructions are rigorously verified, the work supplies a coherent analytic toolkit that could support precise asymptotic analysis in lattice-point problems on non-circular domains, extending the Hardy-type identities from prior work into a systematic fractional-calculus and complex-analytic framework.

major comments (3)

[Introduction] The central constructions presuppose that the functions J^{[p]}_{ω,φ} are already well-defined and sufficiently regular from the Fourier analysis on p-circle domains (Introduction and §2); no re-derivation, uniform-convergence proof, or uniqueness statement for the full parameter range 0 < p ≤ 2 with 2/p ∈ ℕ is supplied in the manuscript, rendering all subsequent fractional-derivative hierarchies, integral representations, and complex extensions dependent on an external reference whose details are not checked here.
[§3] §3 (hierarchical structure): the claim that the Erdélyi-Kober operators produce a well-defined hierarchy requires explicit verification that the resulting functions remain in the same function class and satisfy the original oscillatory integral representation; without error estimates or an inductive step that closes under the operator, the hierarchy is formal rather than analytic.
[§4] §4 (integral representations): the real-analytic formulas are stated to be suitable for axis-dependent asymptotics, yet no explicit error bounds, remainder terms, or comparison with the defining Fourier integral are provided, so it is unclear whether these representations actually recover the original functions or merely satisfy a related integral equation.

minor comments (2)

[§3] Notation for the fractional-order parameters in the Erdélyi-Kober operators should be clarified with respect to the index ω; the current usage risks confusion with the frequency parameter.
[Introduction] The abstract and introduction refer to “previous work” for the oscillatory identities; a single-sentence recap of the defining Fourier integral would improve self-contained readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [Introduction] The central constructions presuppose that the functions J^{[p]}_{ω,φ} are already well-defined and sufficiently regular from the Fourier analysis on p-circle domains (Introduction and §2); no re-derivation, uniform-convergence proof, or uniqueness statement for the full parameter range 0 < p ≤ 2 with 2/p ∈ ℕ is supplied in the manuscript, rendering all subsequent fractional-derivative hierarchies, integral representations, and complex extensions dependent on an external reference whose details are not checked here.

Authors: We agree that greater self-containment would improve accessibility. In the revision we will add to §2 a concise recap of the definition of J^{[p]}_{ω,φ} via Fourier analysis on p-circle domains, together with a statement on regularity and uniqueness for 0 < p ≤ 2 with 2/p natural, citing the prior work for the full proofs. A complete re-derivation is omitted to avoid duplication, but the key convergence properties will be summarized explicitly. revision: partial
Referee: [§3] §3 (hierarchical structure): the claim that the Erdélyi-Kober operators produce a well-defined hierarchy requires explicit verification that the resulting functions remain in the same function class and satisfy the original oscillatory integral representation; without error estimates or an inductive step that closes under the operator, the hierarchy is formal rather than analytic.

Authors: The referee is correct that an explicit inductive argument is required. We will revise §3 to include a proof by induction showing that each application of the Erdélyi-Kober operator maps the class of p-Bessel functions into itself while preserving the oscillatory integral representation, together with the necessary error estimates derived from the operator bounds. revision: yes
Referee: [§4] §4 (integral representations): the real-analytic formulas are stated to be suitable for axis-dependent asymptotics, yet no explicit error bounds, remainder terms, or comparison with the defining Fourier integral are provided, so it is unclear whether these representations actually recover the original functions or merely satisfy a related integral equation.

Authors: We accept this criticism. Section 4 will be expanded to supply explicit error bounds and remainder estimates for the real-analytic integral representations, together with a direct comparison establishing that they recover the original functions defined by the Fourier integrals. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for foundational definition; analytic derivations remain independent

specific steps

self citation load bearing [Abstract]
"While previous work established Hardy-type oscillatory identities for these domains, expressing lattice point discrepancies via p-Bessel functions, the present paper focuses on the intrinsic analytic properties of the functions themselves. In particular, we (i) construct a hierarchical structure of {J^[p]}_{ω,φ} using Erdélyi-Kober-type fractional derivatives, (ii) derive explicit real-analytic integral representations suitable for investigating axis-dependent asymptotic behavior, and (iii) extend the functions to the complex domain through Poisson-type integral formulas."

The three listed constructions presuppose that J^[p]_{ω,φ} are already well-defined and sufficiently regular from the Fourier analysis in the cited previous work. The claim that the results 'establish p-Bessel functions as genuinely new oscillatory kernels' therefore inherits its foundational objects from the self-cited prior definition without re-derivation or independent verification inside this manuscript.

full rationale

The manuscript assumes p-Bessel functions are already defined via Fourier analysis on p-circle domains from prior work and then derives new properties (hierarchical structure via Erdélyi-Kober derivatives, real-analytic integral representations, Poisson-type complex extensions). No step reduces a central claim to a fitted parameter, self-referential definition, or unverified self-citation chain by construction. The reference to previous work supplies the base objects but does not force the new analytic results; those rest on standard fractional calculus and integral transform techniques applied to the given functions. This is ordinary dependence on prior literature rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior existence of the p-Bessel functions from Fourier analysis on p-circle domains; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the functions themselves.

axioms (1)

domain assumption p-Bessel functions exist as defined from Fourier analysis on planar domains bounded by p-circles with 0 < p ≤ 2 and (2/p) natural.
Stated in the abstract as the origin of the functions under study.

pith-pipeline@v0.9.0 · 5494 in / 1287 out tokens · 55475 ms · 2026-05-15T01:40:27.101209+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.

p-Bessel functions defined by outer series (1.1) with inner combinatorial Φ^{[p]}_{k,φ} (1.2); hierarchical structure via Erdélyi-Kober D^γ (Theorem 1.2); integral reps (Theorem 1.3); Poisson extension (Theorem 1.4)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Hardy-type identity (Theorem 1.1) and lattice-point discrepancy P_p(r) expressed via p-Bessel sums

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

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