On an identity for zeros of Bessel functions

Baricz, Árpád; Maširević, Dragana Jankov; Pogány, Tibor K.; Szász, Róbert

doi:10.1016/j.jmaa.2014.08.014

Cited by 14 publications

(7 citation statements)

References 9 publications

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“…Recently we became aware of the paper by Baricz et al [2] in which the authors have pointed out a formula derived in 1977 by Calogero [3] from which we can derive our particular case of the general results by Rayleigh and Sneddon after a few passages. Indeed, Calogero, based on the well known infinite product representation of the Bessel functions of the first kind (4.1)…”

Section: " Resistor Capacitormentioning

confidence: 84%

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On infinite series concerning zeros of Bessel functions of the first kind

Giusti

Mainardi

2016

Eur. Phys. J. Plus

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Abstract. A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the RayleighSneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input.

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Section: " Resistor Capacitormentioning

confidence: 84%

“…On the one hand, the approach discussed in Section 4 is essentially based on a noteworthy formula by Calogero that was motivated by the researches about the connection between the motion of poles and zeros of special solutions of partial differential equations and many-body problems as outlined in [2].…”

Section: Discussionmentioning

confidence: 99%

On infinite series concerning zeros of Bessel functions of the first kind

Giusti

Mainardi

2016

Eur. Phys. J. Plus

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“…As an example, we discuss an extension of the Calogero's formula for the zeros α ν,k of the Bessel function J ν (z) [57] (see also [58,59])…”

Section: Disk and Ballmentioning

confidence: 99%

A physicist's guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

Grebenkov

2019

Preprint

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We summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains. Such spectral sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier-Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review summarizes many summation formulas for spectral sums that are dispersed in the literature.

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“…This question, which has both mathematical and physical interest, has a somehow classical flavour and goes back to lord Rayleigh in 1874 [20] (in fact, the sum (1.3) as a function of α is usually called the Rayleigh function). Since then, many papers have been published which study, with different approaches, these and many other series along with identities and other properties; see, for instance, [23,15,14,3,10] or [19, formula 11 in § 5.7.33] (and the list is by no means exhaustive). However, identities relating (1.3) or any other sum involving the zeros of Bessel functions with some kind of "Bernoulli" or "Euler numbers" seem to be unknown.…”

Section: Introductionmentioning

confidence: 99%