Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory

Arindam Roy; Atul Dixit

arxiv: 1907.03650 · v1 · pith:G3INLOXWnew · submitted 2019-07-08 · 🧮 math.NT · math-ph· math.MP

Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory

Atul Dixit , Arindam Roy This is my paper

Pith reviewed 2026-05-25 01:03 UTC · model grok-4.3

classification 🧮 math.NT math-phmath.MP

keywords Bessel functionsintegral identitiesVoronoi summation formulahypergeometric functionsRamanujan's Lost Notebookmodified Bessel functionsnumber theory

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The pith

A kernel mixing Bessel functions J, K and Y produces an integral identity that, combined with Voronoi summation, transforms series of I and K products into hypergeometric expressions.

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

The paper derives an integral identity that replaces the single Bessel function J_s in earlier Fock-type and Koshliakov integrals with a combination of J_s, K_s and Y_s. This new identity is then inserted into the Voronoi summation formula to obtain a general transformation relating infinite series whose terms are products of modified Bessel functions I_λ(ξ) and K_λ(ξ) to expressions involving the Gaussian hypergeometric function. The transformation is used to recover several explicit evaluations, one of which the authors present as a corrected form of an identity recorded on page 336 of Ramanujan's Lost Notebook. A sympathetic reader cares because the result supplies a new bridge between classical integral representations arising in electromagnetism and summation techniques that appear repeatedly in analytic number theory.

Core claim

We derive an analogous integral identity in which the integrand kernel is a linear combination of J_s(ξ), K_s(ξ) and Y_s(ξ) instead of the single function J_s(ξ) appearing in Koshliakov's generalization of Fock's integral. When this identity is substituted into the Voronoi summation formula, it yields a transformation that equates sums of products I_λ(ξ)K_λ(ξ) to series or closed forms involving the Gaussian hypergeometric function _2F1. Applications of the transformation include a corrected version of the first identity on page 336 of Ramanujan's Lost Notebook.

What carries the argument

The mixed kernel formed by the linear combination of J_s(ξ), K_s(ξ) and Y_s(ξ) that replaces the single J_s(ξ) term and makes the integral identity compatible with Voronoi summation.

If this is right

The transformation equates infinite series of I_λ(ξ)K_λ(ξ) products to expressions containing the Gaussian hypergeometric function.
Several explicit evaluations of such series follow by specializing the transformation.
A corrected form of the identity on page 336 of Ramanujan's Lost Notebook is obtained as one application.
The same kernel identity can be reused with other summation formulas whenever the resulting integral converges.

Where Pith is reading between the lines

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

The same kernel construction may be tested with other classical summation formulas such as Poisson or Abel summation to produce further transformations.
The corrected Ramanujan identity could be checked against independent evaluations obtained from modular-form methods or q-series identities.
If the kernel extends to non-integer orders or complex parameters, the transformation might generate new identities for series appearing in partition theory.

Load-bearing premise

The newly constructed kernel of J_s, K_s and Y_s yields a convergent integral identity that remains valid when inserted into the Voronoi summation formula for the parameter ranges and series considered in the applications.

What would settle it

Numerical agreement, to high precision, between both sides of the derived transformation for concrete admissible values of λ and ξ, or direct verification that the corrected Ramanujan identity on page 336 holds for the stated parameters.

read the original abstract

Closed-form evaluations of certain integrals of $J_{0}(\xi)$, the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann etc. Koshliakov's generalization of one such integral, which contains $J_s(\xi)$ in the integrand, encompasses several important integrals in the literature including Sonine's integral. Here we derive an analogous integral identity where $J_{s}(\xi)$ is replaced by a kernel consisting of a combination of $J_{s}(\xi)$, $K_{s}(\xi)$ and $Y_{s}(\xi)$ that is of utmost importance in number theory. Using this identity and the Vorono\"{\dotlessi} summation formula, we derive a general transformation relating infinite series of products of Bessel functions $I_{\lambda}(\xi)$ and $K_{\lambda}(\xi)$ with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page $336$ of Ramanujan's Lost Notebook.

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

New mixed Bessel kernel integral identity is the core contribution, but its use in Voronoi summation for the Ramanujan correction rests on unshown convergence details.

read the letter

The main new thing here is the integral identity that swaps the usual J_s kernel for a linear combination of J_s, K_s, and Y_s. That combination shows up in some arithmetic applications, so the identity itself could be useful for people working with those kernels. They then feed it into the Voronoi formula to produce a general transformation that turns series of I_λ K_λ products into hypergeometric series, and they apply the transformation to recover or correct a specific Ramanujan entry from the lost notebook. That last step is the part that will interest the narrow audience working on Ramanujan-type identities and Bessel series in number theory. The derivation of the integral looks like a direct analogue of the Fock-Koshliakov line, which is straightforward once the kernel is chosen. The applications are presented as concrete, including the claimed correction. The soft spot is exactly the one flagged in the stress test: the paper needs to show that the new integral converges absolutely enough to justify the interchange with Voronoi for the parameter ranges that appear in the I K series and in the Ramanujan case. Without explicit majorants or dominated-convergence checks, the justification for those steps is not yet visible. If the full write-up supplies those estimates, the gap closes; if it does not, the transformation claim is only formal. The citation pattern looks standard for this subfield and does not lean on self-referential fitting. This is a paper for analytic number theorists who already care about Bessel-function transformations and Ramanujan notebook entries. It is narrow but self-contained enough that a serious referee should see it, mainly to check the convergence details and verify the numerical or formal correctness of the Ramanujan correction. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper derives an analogue of Koshliakov's generalization of Fock-type integrals, replacing the J_s(ξ) integrand with a kernel combining J_s(ξ), K_s(ξ) and Y_s(ξ). It then invokes the Voronoï summation formula to obtain a general transformation relating series of products I_λ(ξ)K_λ(ξ) to expressions involving the Gaussian hypergeometric function, and applies the transformation to produce several identities, including a claimed correction to the first identity on p. 336 of Ramanujan's Lost Notebook.

Significance. If the integral identity and its substitution into Voronoï are rigorously justified, the work would supply a new integral-transform tool for evaluating arithmetic series involving products of modified Bessel functions, extending techniques already used for modular-form coefficients and potentially yielding verifiable corrections to entries in Ramanujan's notebooks.

major comments (2)

[integral identity derivation] The derivation of the integral identity with the combined J_s/K_s/Y_s kernel (abstract and the section presenting the identity) asserts convergence but supplies no explicit majorant, dominated-convergence argument, or growth estimates that would justify absolute convergence and interchange with the Voronoï sum for the ranges of Re(s), λ and ξ appearing in the subsequent applications to I_λ K_λ series.
[applications to series transformations] The applications that invoke the new transformation inside Voronoï (including the claimed correction to Ramanujan's p. 336 identity) presuppose that the kernel integral remains valid under the summation conditions of Voronoï; no separate verification or reference to standard estimates for the oscillatory-plus-exponential kernel is given, rendering the correction claim dependent on an unproven interchange.

minor comments (2)

Notation for the combined kernel is introduced without an explicit displayed equation number, making cross-references in the applications section difficult to follow.
The abstract states that the identity 'encompasses several important integrals' but does not list the recovered classical cases (Sonine, etc.) with equation numbers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major concerns point by point below.

read point-by-point responses

Referee: The derivation of the integral identity with the combined J_s/K_s/Y_s kernel (abstract and the section presenting the identity) asserts convergence but supplies no explicit majorant, dominated-convergence argument, or growth estimates that would justify absolute convergence and interchange with the Voronoï sum for the ranges of Re(s), λ and ξ appearing in the subsequent applications to I_λ K_λ series.

Authors: We agree that the manuscript would benefit from more explicit convergence arguments. In the revised version, we will supply growth estimates for the kernel functions in the relevant ranges of Re(s), λ and ξ, together with a dominated-convergence justification for the integral representation and its interchange with the Voronoï sum. revision: yes
Referee: The applications that invoke the new transformation inside Voronoï (including the claimed correction to Ramanujan's p. 336 identity) presuppose that the kernel integral remains valid under the summation conditions of Voronoï; no separate verification or reference to standard estimates for the oscillatory-plus-exponential kernel is given, rendering the correction claim dependent on an unproven interchange.

Authors: We acknowledge that a dedicated verification of the interchange under the summation conditions of Voronoï is required. In revision we will add a subsection providing asymptotic estimates for the oscillatory-plus-exponential kernel, drawing on standard Bessel-function bounds, to justify the interchange for the series under consideration and thereby support the claimed correction to the Ramanujan identity. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained via external Voronoï formula and novel kernel identity

full rationale

The paper constructs a new integral identity by direct analogy to Koshliakov's known generalization of Fock-type integrals, replacing the J_s integrand with an explicit linear combination of J_s, K_s and Y_s; this step is presented as an independent derivation rather than a fit or renaming. The subsequent transformation is obtained by substituting the identity into the standard Voronoï summation formula (an externally established arithmetic tool with no author overlap cited as load-bearing). No equation reduces to a self-defined quantity, no parameter is fitted then relabeled as prediction, and no uniqueness theorem or ansatz is smuggled via self-citation. The Ramanujan correction and hypergeometric series relations follow from these steps without circular reduction. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of Bessel functions, convergence of the defining integrals, and applicability of the Voronoï summation formula; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

standard math Standard analytic properties and integral representations of Bessel functions J, Y, K, I hold in the relevant parameter ranges.
Invoked implicitly when constructing the new kernel and applying the identity.
domain assumption Voronoï summation formula applies directly to the series of Bessel products under consideration.
Central step in obtaining the general transformation.

pith-pipeline@v0.9.0 · 5755 in / 1357 out tokens · 23509 ms · 2026-05-25T01:03:32.333073+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

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Abramowitz and I

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 9th edition, Dover publications, New York, 1970

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B. C. Berndt, Identities involving the coeﬃcients of a class of Dirichlet series, III , Trans. Amer. Math. Soc. 146 (1969), 323–348

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B. C. Berndt, O.-Y. Chan, S.-G. Lim and A. Zaharescu, Questionable claims found in Ramanujan ’s lost notebook, in Tapas in Experimental Mathematics , T. Amdeberhan and V. H. Moll, eds., Contemp. Math., Vol. 457, American Mathematical Society, Providence, RI, 2 008, pp. 69–98

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B. C. Berndt, A. Dixit, S. Kim and A. Zaharescu, Sums of squares and products of Bessel functions , Adv. Math. 338 (2018), 305–338

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B. C. Berndt, A. Dixit, A. Roy and A. Zaharescu, New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan , Adv. Math. 304 (2017), 809–929

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B. C. Berndt, S. Kim and A. Zaharescu, The circle problem of Gauss and the divisor problem of Dirich let - still unsolved , Amer. Math. Monthly 125 No. 2 (2018), 99–114

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Dixit and V

A. Dixit and V. H. Moll, Self-reciprocal functions, powers of the Riemann zeta func tion and modular-type transformations, J. Number Theory 147 (2015), 211–249

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Dixit, N

A. Dixit, N. Robles, A. Roy and A. Zaharescu, Koshliakov kernel and identities involving the Riemann zeta function , J. Math. Anal. Appl. 435 No. 2 (2016), 1107–1128

work page 2016

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Dixon and W.L

A.L. Dixon and W.L. Ferrar, Some summations over the lattice points of a circle (I), Quart. J. Math. 1 (1934), 48–63

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A. L. Dixon and W. L. Ferrar, Inﬁnite integrals of Bessel functions , Quart. J. Math. 1 (1935), 161–174

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A. L. Dixon and W. L. Ferrar, Inﬁnite integrals in the theory of Bessel functions , Quart. J. Math. 1 (1930), 122–145

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T. M. Dunster, Bessel functions of purely imaginary order, with an applica tion to second-order linear diﬀerential equations having a large parameter , SIAM. J. Math. Anal. 21 No. 4 (1990), 995–1018. ANALOGUE OF A FOCK-TYPE INTEGRAL ARISING FROM ELECTROMAGNE TISM 29

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Endres and F

S. Endres and F. Steiner, A simple inﬁnite quantum graph , Ulmer Seminare Funktionalanalysis und Diﬀerentialgleichungen 14 (2009), 187–200

work page 2009

[15] [15]

Egger and F

S. Egger and F. Steiner, An exact trace formula and zeta functions for an inﬁnite quan tum graph with a non-standard Weyl asymptotics , J. Phys. A: Math. Theor. 44, No. 44 (2011), 185–202

work page 2011

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V. A. Fock, Zur Berechnung des elektromagnetischen Wechselstromfelde s bei ebener Begrenzung , Ann. Phys. 409 (4) (1933), 401–420

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V. A. Fock and V. R. Bursian, Electromagnetic ﬁeld of alternating current in a circuit wi th two groundings (in Russian), J. Russian Phys.-Chem. Soc. (Zhurnal Russkag o Fiziko-Khimicheskago Obshchestva) 58 No. 2 (1926), 355–363. (Available at http://books.e-heritage.ru/book/10084021)

work page arXiv 1926

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A. P. Guinand, Summation formulae and self-reciprocal functions (II), Quart. J. Math. 10 (1939), 104– 118

work page 1939

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A. P. Guinand, Some rapidly convergent series for the Riemann ξ-function, Quart.J. Math. (Oxford) 6 (1955), 156–160

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A. P. Guinand, Concordance and the harmonic analysis of sequences , Acta Math. 101, No. 3 (1959), 235–271

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I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products , 7th ed., Academic Press, San Diego, 2007

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G. H. Hardy, On the expression of a number as the sum of two squares , Quart. J. Pure Appl. Math. 46 (1915), 263–283

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G. H. Hardy, On Dirichlet’s divisor problem , Proc. London Math. Soc. (2) 15 (1916), 1–25

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N. S. Koshliakov, On a certain deﬁnite integral connected with the cylindric f unction Jµ(x), C. R. Acad. Sci. URSS 2 (1934), 145–147

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N. S. Koshliakov, On Voronoi’s sum-formula , Mess. Math. 58 (1929), 30–32

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N. S. Koshliakov, Note on certain integrals involving Bessel functions , Bull. Acad. Sci. URSS Ser. Math. 2 No. 4, 417–420; English text 421–425 (1938)

work page 1938

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Landau, Vorlesungen ¨ uber Zahlentheorie, Bd

E. Landau, Vorlesungen ¨ uber Zahlentheorie, Bd. 1, S. Hirzel, Leipzig, 1927

work page 1927

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A. P. Lursmanaˇsvili, On the number of lattice points in multidimensional spheres (Russian), Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze 19, (1953), 79–120

work page 1953

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A. P. Lursmanaˇsvili, On the number of lattice points in multidimensional spheres of odd dimension (Geor- gian), Soobˇsˇceniya Akad. Nauk Gruzin. SSR. 14 (1953), 513–520

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A. I. Popov, On some summation formulas (in Russian), Bull. Acad. Sci. L’URSS, 7 (1934), 801–802

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