The zeta function constructed from the zeros of the Bessel function

Actor, Alfred; Bender, I.

doi:10.1088/0305-4470/29/20/013

Cited by 13 publications

(7 citation statements)

References 13 publications

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“…A natural question, left open here, is whether this generalization of Riemann's zeta function satisfies a functional equation. See also [55, §15.51], [2] and [62, §11.6] regarding the generalization of the Riemann zeta function obtained by consideration of the sum (75) with the factor J 2 ν+1 (j ν,n ) replaced by 1. By comparison of (70) and (74) we obtain a sequence of evaluations of Σ(ν, q) for q = 2n−2ν > 0 and n = 0, 1, .…”

Section: Momentsmentioning

confidence: 99%

The Law of the Maximum of a Bessel Bridge

Pitman

Yor

1999

Electron. J. Probab.

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Let M δ be the maximum of a standard Bessel bridge of dimension δ. A series formula for P (M δ ≤ a) due to Gikhman and Kiefer for δ = 1, 2, . . . is shown to be valid for all real δ > 0. Various other characterizations of the distribution of M δ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of M δ is described both as δ → ∞ and as δ ↓ 0.

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Section: Momentsmentioning

confidence: 99%

The Law of the Maximum of a Bessel Bridge

Pitman

Yor

1999

Electron. J. Probab.

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“…It was thus dubbed the Bessel zeta function in Ref. [21,22], it is also known as the Raleigh function [23], for even integers s = 2, 4, 6, . .…”

Section: Twist Field As Aharonov-bohm Vortexmentioning

confidence: 99%

Vacuum expectation value of twist fields

Belitsky

2017

Phys. Rev. D

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Twist fields emerge in a number of physical applications ranging from entanglement entropy to scattering amplitudes in four-dimensional gauge theories. In this work, their vacuum expectation values are studied in the path integral framework. By performing a gauge transformation, their correlation functions are reduced to field theory of matter fields in external AharonovBohm vortices. The resulting functional determinants are then analyzed within the zeta function regularization for the spectrum of Bessel zeros and concise formulas are derived.

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“…Convolutions of Rayleigh functions with respect to the power l were considered in [1,3,4,7,10,11]. N. Meiman [10] was the first to obtain a compact recursion formula σ l (ν) = 1 ν + l l−1 k=1 σ l−k (ν)σ k (ν) for l = 2, 3, 4, .…”

Section: Introductionmentioning

confidence: 99%

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Варламов¹

2007

Journal of Mathematical Analysis and Applications

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Rayleigh functions σ l (ν) are defined as series in inverse powers of the Bessel function zeros λ ν,n = 0, σ l (ν) = ∞ n=1 1 λ 2l ν, n , where l = 1, 2, . . . ; ν is the index of the Bessel function J ν (x) and n = 1, 2, . . . is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l (m) = ∞ k=−∞ σ l |m − k| σ l |k| for l = 1, 2, . . . ; m = 0, ±1, ±2, . . . , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699-725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424], where the properties of R 1 (m) were investigated.In the present work a general representation of R l (m) in terms of σ l (ν) is deduced. On the basis of this a representation for the function R 2 (m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 (m) as |m| → ∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded ✩ domains with periodicity conditions in one coordinate. As an example of application of R l (m) a forced Boussinesq equation u tt − 2bΔu t = −αΔ 2 u + Δu + βΔ u 2 + f with α, b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 (m) and R 2 (m) which are responsible for the nonlinear smoothing effect.

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The zeta function constructed from the zeros of the Bessel function

Cited by 13 publications

References 13 publications

The Law of the Maximum of a Bessel Bridge

The Law of the Maximum of a Bessel Bridge

Vacuum expectation value of twist fields

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

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