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arxiv: 2605.13882 · v1 · pith:DPZZESCSnew · submitted 2026-05-11 · 🧮 math.NT

A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function

S. A. Dar , R. P. Paris This is my paper

Pith reviewed 2026-05-15 05:57 UTC · model grok-4.3

classification 🧮 math.NT MSC 33C60
keywords Ramanujan integralMeijer G-functionMellin-Barnes contourFourier cosine transforminfinite seriesclosed form
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The pith

The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.

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

This paper derives analytical expressions for the Ramanujan integral R_C(m,n), which integrates x^m times cos(π n x) over exp(2π √x) minus one from zero to infinity. The derivation uses Mellin-Barnes contour representations of the cosine to convert the integral into a series of Meijer G-functions. Generalizations of the integral receive the same treatment and also reduce to Meijer G series. These results supply closed forms for nine infinite series of Meijer G-functions as a direct application. A reader cares because the expressions replace numerical quadrature with explicit series that can be studied or computed in analytic number theory.

Core claim

The integral R_C(m,n) equals an infinite series of Meijer G-functions of one variable, obtained by substituting the Mellin-Barnes contour representation of the cosine and integrating term by term under stated convergence conditions on m and n. The same contour method produces analogous series expressions for the generalized integrals I_C^*(υ,b,c,λ,y), Ξ_C(υ,b,c,λ,y), ∇_C(υ,b,c,λ,y) and I_C(υ,b,λ,y). Application of the R_C(m,n) result then supplies explicit closed-form evaluations for nine infinite series of Meijer G-functions.

What carries the argument

Mellin-Barnes contour integral representation of the cosine function, which converts the oscillatory integral into a form that integrates against the remaining factors to produce Meijer G-functions.

If this is right

  • The integral R_C(m,n) possesses an explicit series representation whenever the convergence conditions hold.
  • The four listed generalizations of R_C(m,n) likewise admit expressions as infinite series of Meijer G-functions.
  • Nine infinite series of Meijer G-functions acquire closed-form evaluations through direct substitution of the R_C(m,n) result.
  • The method supplies a uniform contour-integral route to analytic continuation of these integrals beyond the original domain of convergence.

Where Pith is reading between the lines

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

  • The same contour technique may apply to other Fourier transforms whose kernels involve Bose-Einstein-type denominators.
  • The resulting series could be used to extract asymptotic information or modular properties for special values of the parameters.
  • Direct numerical summation of the Meijer G series for concrete m and n offers an independent check on the analytic continuation formulas.

Load-bearing premise

Interchange of integration order is valid and the resulting contour integrals converge absolutely for the stated ranges of m and n.

What would settle it

For m = 1/2 and n = 1, compute the numerical value of the integral to high precision and compare it with the partial sums of the derived Meijer G series; a discrepancy larger than truncation error falsifies the claimed equality.

read the original abstract

In this paper, we obtain analytical evaluations of the Ramanujan integral \[\textbf{R}_{C}(m,n)= \int_{0}^{\infty}\frac{x^m\,\cos(\pi nx)}{\exp{(2\pi\sqrt{x})-1}}dx\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\upsilon,b,c,\lambda,y)$ ,$\Xi_{C}(\upsilon,b,c,\lambda,y)$, $\nabla_{C}(\upsilon,b,c,\lambda,y)$ and $I_{C}(\upsilon,b,\lambda,y)$. These integrals are also expressed in terms of infinite series of Meijer $G$-functions. Moreover, as an application of a Ramanujan's integral $\textbf{R}_C(m,n)$, the closed-form evaluations of nine infinite series of Meijer $G$-functions are obtained.

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

2 major / 2 minor

Summary. The paper claims to obtain analytical evaluations of the Ramanujan integral R_C(m,n) = ∫_0^∞ [x^m cos(π n x) / (exp(2π √x) - 1)] dx as an infinite series of Meijer G-functions of one variable, via Mellin-Barnes contour representations of the cosine, subject to suitable convergence conditions on m and n. It further treats four generalizations (I_C^*, Ξ_C, ∇_C, I_C) in the same form and applies the main result to derive closed forms for nine infinite series of Meijer G-functions.

Significance. If the contour-integral derivations can be made rigorous, the work would supply explicit Meijer-G series representations for a family of Ramanujan-type Fourier cosine integrals, extending classical contour techniques to this setting and yielding concrete evaluations of related G-function series that may be of interest in analytic number theory and special-function identities.

major comments (2)
  1. [Main derivation of R_C(m,n) (contour-integral step)] The central derivation (substitution of the Mellin-Barnes representation of cos(π n x) into R_C(m,n), followed by interchange of the x-integral and contour integral to produce the Meijer-G series) is presented formally but without explicit justification. No Stirling-type growth bounds on |Γ(s)Γ(1/2-s) x^{-s}| for Re(s)=c and |Im(s)|→∞, nor a dominated-convergence argument controlling the integrand uniformly in the contour parameter, are supplied to validate the interchange or the absolute convergence of the resulting double integral under the stated conditions on m and n.
  2. [Series representation of R_C(m,n)] The absolute convergence of the infinite series of Meijer G-functions obtained after residue evaluation is asserted inside the claimed region for m and n, but no explicit radius-of-convergence estimates, comparison tests, or asymptotic analysis of the G-function terms are provided to confirm this.
minor comments (2)
  1. [Generalizations section] The precise parameter domains for the four generalized integrals I_C^*, Ξ_C, ∇_C and I_C are stated only qualitatively; explicit inequalities relating υ, b, c, λ, y to the original m, n would clarify the scope.
  2. [Application section] A few typographical inconsistencies appear in the notation for the Meijer G-function arguments across the main theorem and the application to the nine series.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the rigor of our derivations. We agree that explicit justifications for the contour interchange and series convergence strengthen the manuscript and have incorporated the necessary details in the revision.

read point-by-point responses

  1. Referee: The central derivation (substitution of the Mellin-Barnes representation of cos(π n x) into R_C(m,n), followed by interchange of the x-integral and contour integral to produce the Meijer-G series) is presented formally but without explicit justification. No Stirling-type growth bounds on |Γ(s)Γ(1/2-s) x^{-s}| for Re(s)=c and |Im(s)|→∞, nor a dominated-convergence argument controlling the integrand uniformly in the contour parameter, are supplied to validate the interchange or the absolute convergence of the resulting double integral under the stated conditions on m and n.

    Authors: We appreciate this observation. In the revised manuscript we have added a dedicated subsection justifying the interchange. Using Stirling's approximation we establish that |Γ(s)Γ(1/2-s)| decays exponentially as |Im(s)|→∞ on the vertical line Re(s)=c, yielding an integrable majorant (uniform in the contour parameter) of the form C x^{Re(m)-c} / |exp(2π√x)-1| times an exponentially decaying factor. The dominated convergence theorem then permits the interchange under the stated conditions Re(m)>-1 and |n|<1. These bounds are now stated explicitly. revision: yes

  2. Referee: The absolute convergence of the infinite series of Meijer G-functions obtained after residue evaluation is asserted inside the claimed region for m and n, but no explicit radius-of-convergence estimates, comparison tests, or asymptotic analysis of the G-function terms are provided to confirm this.

    Authors: We acknowledge the omission. The revised version includes an asymptotic analysis of the general term of the Meijer-G series for large index k. Employing standard asymptotic expansions of G-functions with fixed argument, we obtain that the k-th term is O(r^k k^β) with |r|<1 inside the region |n|<1, Re(m)>-1. The ratio test then confirms absolute convergence. These estimates appear in the new Section 4.2. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Mellin-Barnes substitution yields Meijer-G series without self-referential reduction

full rationale

The derivation begins with the given integral R_C(m,n) and substitutes the known Mellin-Barnes contour representation of cos(π n x), which is an external identity independent of the target result. After formal interchange (under stated convergence conditions) and residue evaluation, the outcome is expressed as an infinite series of Meijer G-functions. Meijer G is a pre-existing special function defined by its own contour integral; representing the Ramanujan integral in terms of it does not redefine or presuppose the result. No fitted parameters, self-citations as load-bearing premises, or ansatz smuggled from prior work by the same authors appear in the chain. The steps remain self-contained against external Mellin-Barnes and Meijer-G identities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on the standard Mellin-Barnes representation of the cosine and the definition of the Meijer G-function; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math Mellin-Barnes contour integral representation of the cosine function
    Invoked to convert the cosine into a contour integral that permits interchange with the remaining integrand.

pith-pipeline@v0.9.0 · 5502 in / 1209 out tokens · 45469 ms · 2026-05-15T05:57:50.950038+00:00 · methodology

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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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    Relation between the paper passage and the cited Recognition theorem.

    We obtain analytical evaluations of the Ramanujan integral RC(m,n) ... in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
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extends
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The paper appears to rely on the theorem as machinery.
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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

20 extracted references · 20 canonical work pages

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