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Zeros of Bessel function derivatives


Authors: Árpád Baricz, Chrysi G. Kokologiannaki and Tibor K. Pogány
Journal: Proc. Amer. Math. Soc. 146 (2018), 209-222
MSC (2010): Primary 33C10, 30D15
DOI: https://doi.org/10.1090/proc/13725
Published electronically: August 1, 2017
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Abstract: We prove that for $ \nu >n-1$ all zeros of the $ n$th derivative of the Bessel function of the first kind $ J_{\nu }$ are real. Moreover, we show that the positive zeros of the $ n$th and $ (n+1)$th derivative of the Bessel function of the first kind $ J_{\nu }$ are interlacing when $ \nu \geq n$ and $ n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $ n$th derivative, properties of the Laguerre-Pólya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivatives of the Struve function of the first kind $ \mathbf {H}_{\nu }$ are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel and Struve functions of the first kind. Some open problems related to Hurwitz's theorem on the zeros of Bessel functions are also proposed.


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Additional Information

Árpád Baricz
Affiliation: Department of Economics, Babeş-Bolyai University, 400591 Cluj-Napoca, Romania; Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary
Email: bariczocsi@yahoo.com

Chrysi G. Kokologiannaki
Affiliation: Department of Mathematics, University of Patras, 26500 Patras, Greece
Email: chrykok@math.upatras.gr

Tibor K. Pogány
Affiliation: Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia; Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary
Email: poganj@pfri.hr

DOI: https://doi.org/10.1090/proc/13725
Keywords: Zeros of Bessel and Struve functions, Laguerre-P\'olya class of entire functions, interlacing of positive zeros, reality of the zeros, Laguerre inequality, Jensen polynomials, Laguerre polynomials, Rayleigh sums
Received by editor(s): December 13, 2016
Received by editor(s) in revised form: January 31, 2017, and February 16, 2017
Published electronically: August 1, 2017
Communicated by: Mourad E. H. Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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