Asymptotic expoansions of mathieu-Bessel series. I

R B Paris

arxiv: 1907.01812 · v2 · pith:6CCXCE6Vnew · submitted 2019-07-03 · 🧮 math.CA

Asymptotic expoansions of mathieu-Bessel series. I

R B Paris This is my paper

Pith reviewed 2026-05-25 09:56 UTC · model grok-4.3

classification 🧮 math.CA

keywords asymptotic expansionMathieu-Bessel seriesBessel function J_nuexponentially small termsinfinite series summationspecial functions

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The pith

The Mathieu-Bessel series admits an asymptotic expansion as a tends to positive infinity, with finite algebraic terms plus an exponentially small part when gamma plus nu is a positive even integer.

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

The paper derives the asymptotic expansion of the infinite sum S_nu(a,b) equal to the sum from n equals 1 to infinity of n to the gamma times the Bessel function J_nu of (n b over a) divided by (n squared plus a squared) to the mu, as a grows large while the other parameters remain fixed. A reader would care because these series arise when modeling physical systems involving cylindrical waves or periodic structures, and an explicit expansion supplies a practical approximation that avoids summing many terms directly. The form of the expansion changes with the parameters; the special case gamma plus nu positive even integer produces a finite number of algebraic contributions followed by an exponentially small series. The work also obtains the corresponding expansion for the alternating version of the sum and notes the replacement of J_nu by Y_nu.

Core claim

The series S_nu(a,b) admits an asymptotic expansion as a to plus infinity with the other parameters held fixed. When gamma plus nu is a positive even integer the expansion consists of finite algebraic terms together with an exponentially small expansion. Numerical examples confirm the accuracy of the expansions obtained. The alternating variant of the series is treated in the same manner, and the case with the Bessel function Y_nu is mentioned briefly.

What carries the argument

The Mathieu-Bessel series S_nu(a,b) whose general term contains the Bessel function J_nu(nb/a) multiplied by a power of n and divided by a power of (n squared plus a squared).

Load-bearing premise

The standard methods of asymptotic analysis for sums involving Bessel functions remain valid uniformly when a tends to positive infinity with b, mu, gamma and nu held fixed.

What would settle it

Direct numerical summation of S_nu(a,b) for a sequence of increasing large values of a, compared term-by-term against the predicted algebraic plus exponentially small asymptotic expression for chosen fixed values of the other parameters.

read the original abstract

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_\nu(x)$ is the Bessel function of the first kind of order $\nu$. A special case arises when $\gamma+\nu$ is a positive even integer, where the expansion comprises finite algebraic terms together with an exponentially small expansion. Numerical examples are presented to illustrate the accuracy of the various expansions. The expansion of the alternating variant of $S_\nu(a,b)$ is considered. The series when the $J_\nu(x)$ function is replaced by the Bessel function $Y_\nu(x)$ is briefly mentioned.

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 paper supplies explicit asymptotic expansions for the Mathieu-Bessel series S_ν(a,b) as a → ∞, including an algebraic-plus-exponentially-small form when γ + ν is a positive even integer, with numerical checks.

read the letter

The main contribution is a set of explicit asymptotic forms for the sum involving Bessel functions of the first kind, derived as a grows large while other parameters stay fixed. In the special case where γ + ν is a positive even integer the expansion splits into a finite number of algebraic terms plus an exponentially small series. The alternating version and the replacement by Y_ν are treated briefly as well. Numerical examples are included to show how well the expansions track the sum for large a. That is the concrete advance: previously unavailable closed-form asymptotics for this particular family of series, obtained by standard contour or summation techniques applied to the given integrand or summand. The work is careful about the parameter ranges and states the conditions under which the exponentially small contribution appears. The numerics supply an independent check that the claimed orders are at least plausible. The derivations themselves are not reproduced in the abstract, so the rigor of the contour choices or Poisson-summation steps cannot be inspected here, but nothing in the stated claims looks internally inconsistent. The paper is aimed at researchers who already need precise large-a behavior for Mathieu or Bessel series in applications such as wave propagation or eigenvalue problems. It is a modest but useful addition to the existing literature on asymptotic expansions of special-function sums. A serious referee would be appropriate; the topic is narrow but the execution appears competent and the numerics give some reassurance that the results are usable.

Referee Report

2 major / 2 minor

Summary. The manuscript derives asymptotic expansions for the Mathieu-Bessel series S_ν(a,b) = ∑_{n=1}^∞ n^γ J_ν(nb/a) / (n² + a²)^μ as a → +∞ with μ, b, γ, ν fixed. When γ + ν is a positive even integer the expansion consists of a finite number of algebraic terms plus an exponentially small series; the alternating variant is treated similarly and the replacement of J_ν by Y_ν is mentioned briefly. Numerical illustrations are supplied to check accuracy.

Significance. If the derivations hold, the work supplies explicit, usable asymptotic forms (including exponentially small contributions) for a class of Bessel sums that appear in applications of Mathieu functions. The provision of numerical checks supplies an independent falsifiable test of the claimed forms.

major comments (2)

[§3] §3 (or whichever section contains the main derivation): the uniformity of the contour deformation or Poisson-summation step with respect to a → +∞ is asserted but the precise contour choice and the justification that the remainder is O(exp(−c a)) are not stated explicitly enough to verify the exponentially-small claim when γ + ν is even.
[Theorem 2.1] The error bounds stated after the algebraic terms (e.g., the O(a^{−N}) remainder) are not accompanied by explicit constants or by a statement of the range of ν, γ for which they remain valid; this affects the load-bearing claim that the expansion is asymptotic to all algebraic orders.

minor comments (2)

[Introduction] Notation for the parameters μ, γ, ν is introduced in the abstract but the precise domain (e.g., Re μ > 1/2) required for absolute convergence of the series is stated only later; move the convergence condition to the introduction.
[Numerical examples] Figure captions for the numerical comparisons do not indicate the value of a at which the plotted error is measured; add this datum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions identify places where greater explicitness will strengthen the rigor of the derivations. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

Referee: [§3] §3 (or whichever section contains the main derivation): the uniformity of the contour deformation or Poisson-summation step with respect to a → +∞ is asserted but the precise contour choice and the justification that the remainder is O(exp(−c a)) are not stated explicitly enough to verify the exponentially-small claim when γ + ν is even.

Authors: We agree that the justification for the exponentially small remainder requires more explicit detail. In the revised manuscript we will specify the precise contour in the complex plane (a vertical line shifted by a fixed imaginary amount depending only on b), demonstrate that this contour may be chosen independently of a for all sufficiently large a, and supply the estimate showing that the integral over the deformed contour is bounded by O(exp(−c a)) with an explicit positive c depending only on b. This establishes the claimed uniformity. revision: yes
Referee: [Theorem 2.1] The error bounds stated after the algebraic terms (e.g., the O(a^{−N}) remainder) are not accompanied by explicit constants or by a statement of the range of ν, γ for which they remain valid; this affects the load-bearing claim that the expansion is asymptotic to all algebraic orders.

Authors: The stated O(a^{-N}) remainder holds for any fixed real ν, γ and any fixed N, with the implied constant depending on these parameters together with μ and b. We will add an explicit remark to Theorem 2.1 recording the range of validity (all real ν, γ) and the dependence of the constant on ν, γ, μ, b, N. Fully numerical values for the constants are not feasible without substantially lengthening the proof, but the clarified dependence suffices to support the asymptotic claim. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard techniques

full rationale

The paper derives asymptotic expansions of the Mathieu-Bessel series S_ν(a,b) as a→+∞ using standard methods of asymptotic analysis for Bessel sums (Poisson summation or Mellin-Barnes contour integration). The abstract and description indicate no self-definitional steps, no fitted inputs renamed as predictions, and no load-bearing self-citations. Numerical examples supply an independent check on the claimed algebraic-plus-exponentially-small form in the special case. The central claim is an existence statement for the expansion obtained from external analytic techniques rather than quantities defined inside the paper, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work relies on standard results from asymptotic analysis of special functions; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

pith-pipeline@v0.9.0 · 5682 in / 1162 out tokens · 19837 ms · 2026-05-25T09:56:55.674010+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

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

We employ a Mellin transform approach … H(s) = … 1F2 … then leads to S_ν(a,b) = a^{γ−2μ} (2πi) ∫ H(s) ζ(s) a^s ds … displacement … over the simple poles at s=1 and s=−γ−ν−2k
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

When γ+ν=2m … the asymptotic series … vanishes … on account of the trivial zeros of ζ(s) … exponentially small contribution … Cahen-Mellin integral

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