Properties of the Product of Modified Bessel Functions

Baricz, Árpád; Pogány, Tibor K.

doi:10.1007/978-1-4939-0258-3_31

Cited by 5 publications

(2 citation statements)

References 24 publications

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“…D. The need for efficient functional or uniform bounding inequalities for Bessel-type special functions frequently occur in convergence issues regarding the so-called Fourier-Bessel series of the Neumann, Kapteyn, Schlömilch and Dini kind, see [16,24]. However, in applications conditioned with numerical approximations which occur in numerical modeling in the STEM disciplines, almost all Bessel and alike functions' arguments are real, consult B.…”

Section: Discussion: Further Research Goals and Open Problemsmentioning

confidence: 99%

Integral Representations for Products of Two Bessel or Modified Bessel Functions

Maširević

Pogány

2019

Mathematics

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The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments. Keywords: Bessel function of the first kind; modified Bessel function of the first kind; integral representation of product functions; Chebyshev polynomial of the first kind MSC: Primary: 33C10; Secondary: 33C05; 33C20 √ X 2 e it + x 2 e −it , while Nicholson proved that ([5], p. 236, Equation (39)) J n (z)J −n (z) = 1 π sin(nπ) π 0 sin(2nx) J 0 (2z sin x) dx, where z ∈ R and n / ∈ Z. This result implies for n non-negative integer the formula ([5], p. 236, Equation (40)) J 2 n (z) = 1 π π 0 sin(2nx) J 0 (2z sin x) dx .Nicholson also derived integral representation for the mixed product J n (z)Y n (z) of the Bessel function of the first kind J n (z) and of the Bessel function of the second kind Y n (z) when n ∈ Z; similar results can be found also in the already cited papers by Dixon and Ferrar [4,6] and by Görlich et al. [7].

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Section: Discussion: Further Research Goals and Open Problemsmentioning

confidence: 99%

Integral Representations for Products of Two Bessel or Modified Bessel Functions

Maširević

Pogány

2019

Mathematics

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“…ν have been studied by Baricz, Jankov and Pogány [5], while Neumann series of the second type were considered by Baricz and Pogány in somewhat different purposes in [6,7]; see also [31]. An important role has throughout of this paper the Sonin-Gubler formula which connects modified Bessel function of the first kind I ν , modified Struve function L ν and a definite integral of the Bessel function of the first kind J ν [23, p. 424] (actually a special case of a Sonin-formula [63, p. 434]):…”

Section: Introductionmentioning

confidence: 99%

Integral representations and summations of the modified Struve function

Baricz

Pogány

2013

Acta Math Hung

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It is known that Struve function Hν and modified Struve function Lν are closely connected to Bessel function of the first kind Jν and to modified Bessel function of the first kind Iν and possess representations through higher transcendental functions like generalized hypergeometric 1 F 2 and Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for Lν (x). In this paper firstly, we obtain various another type integral representation formulae for Lν (x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by Iν (x) and Lν (x) which are connected by a Sonin-Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich-Wagner line integral expressions are derived for the Bessel function of the first kind Jν and for an associated generalized Schlömilch series.

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Second Type Neumann Series Related to Nicholson’s and to Dixon–Ferrar Formula

Cvijović¹,

Pogány

2020

Trends in Mathematics

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Properties of the Product of Modified Bessel Functions

Cited by 5 publications

References 24 publications

Integral Representations for Products of Two Bessel or Modified Bessel Functions

Integral Representations for Products of Two Bessel or Modified Bessel Functions

Integral representations and summations of the modified Struve function

Second Type Neumann Series Related to Nicholson’s and to Dixon–Ferrar Formula

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