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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
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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  • [AM] R. Álvarez-Nodarse and F. Marcellán, A generalization of the classical Laguerre polynomials, Rend. Circ. Mat. Palermo (2) 44 (1995), no. 2, 315-329. MR 1355489,
  • [BCD] Árpád Baricz, Murat Çağlar, and Erhan Deniz, Starlikeness of Bessel functions and their derivations, Math. Inequal. Appl. 19 (2016), no. 2, 439-449. MR 3458758,
  • [BPS] Árpád Baricz, Saminathan Ponnusamy, and Sanjeev Singh, Turán type inequalities for Struve functions, J. Math. Anal. Appl. 445 (2017), no. 1, 971-984. MR 3543805,
  • [BS] Árpád Baricz and Róbert Szász, Close-to-convexity of some special functions and their derivatives, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 1, 427-437. MR 3439869,
  • [BY] Árpád Baricz and Nihat Yağmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. 42 (2017), no. 2, 325-346. MR 3596935,
  • [Bo] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [CVV] George Csordas, Richard S. Varga, and István Vincze, Jensen polynomials with applications to the Riemann $ \zeta $-function, J. Math. Anal. Appl. 153 (1990), no. 1, 112-135. MR 1080122,
  • [CW] George Csordas and Jack Williamson, The zeros of Jensen polynomials are simple, Proc. Amer. Math. Soc. 49 (1975), 263-264. MR 0361017,
  • [DC] Dimitar K. Dimitrov and Youssèf Ben Cheikh, Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions, J. Comput. Appl. Math. 233 (2009), no. 3, 703-707. MR 2583006,
  • [Hu] A. Hurwitz, Ueber die Nullstellen der Bessel'schen Function, Math. Ann. 33 (1888), no. 2, 246-266 (German). MR 1510541,
  • [IM] Mourad E. H. Ismail and Martin E. Muldoon, Bounds for the small real and purely imaginary zeros of Bessel and related functions, Methods Appl. Anal. 2 (1995), no. 1, 1-21. MR 1337450,
  • [Je] J. L. W. V. Jensen, Recherches sur la théorie des équations, Acta Math. 36 (1913), no. 1, 181-195 (French). MR 1555086,
  • [Ke] M. K. Kerimov, Studies on the zeros of Bessel functions and methods for their computation, Comput. Math. Math. Phys. 54 (2014), no. 9, 1337-1388. MR 3258608,
  • [KK] Haseo Ki and Young-One Kim, On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), no. 1, 45-73. MR 1769725,
  • [Ko] Roelof Koekoek, Generalizations of the classical Laguerre polynomials and some q-analogues, ProQuest LLC, Ann Arbor, MI. Thesis (Dr.)-Technische Universiteit Delft (The Netherlands) (1990). MR 2714461
  • [KL] Chrysi G. Kokologiannaki and Andrea Laforgia, Simple proofs of classical results on zeros of $ J_\nu (x)$ and $ J'_\nu (x)$, Tbilisi Math. J. 7 (2014), no. 2, 35-39. MR 3313053,
  • [KP] Chrysi G. Kokologiannaki and Eugenia N. Petropoulou, On the zeros of $ J_\nu '''(x)$, Integral Transforms Spec. Funct. 24 (2013), no. 7, 540-547. MR 3171971,
  • [Le] B. Ya. Levin, Lectures on entire functions, translated by V. Tkachenko, with written in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996 (Russian). MR 1400006
  • [LM] Lee Lorch and Martin E. Muldoon, The real zeros of the derivatives of cylinder functions of negative order, with dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III, Methods Appl. Anal. 6 (1999), no. 3, 317-326. MR 1803312,
  • [LP] Lee Lorch and Peter Szego, Monotonicity of the zeros of the third derivative of Bessel functions, Methods Appl. Anal. 2 (1995), no. 1, 103-111. MR 1337456,
  • [Me] A. McD. Mercer, The zeros of $ az^2J''_\nu (z)+bzJ'_\nu (z)+cJ_\nu (z)$ as functions of order, Internat. J. Math. Math. Sci. 15 (1992), no. 2, 319-322. MR 1155524,
  • [OLBC] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, with 1 CD-ROM (Windows, Macintosh and UNIX), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. MR 2723248
  • [Ske] E. A. Skelton, A new identity for the infinite product of zeros of Bessel functions of the first kind or their derivatives, J. Math. Anal. Appl. 267 (2002), no. 1, 338-344. MR 1886832,
  • [Sko] H. Skovgaard, On inequalities of the Turán type, Math. Scand. 2 (1954), 65-73. MR 0063415,
  • [St] J. Steinig, The real zeros of Struve's function, SIAM J. Math. Anal. 1 (1970), 365-375. MR 0267162,
  • [Wa] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
  • [WL] R. Wong and T. Lang, On the points of inflection of Bessel functions of positive order. II, Canad. J. Math. 43 (1991), no. 3, 628-651. MR 1118013,

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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

Chrysi G. Kokologiannaki
Affiliation: Department of Mathematics, University of Patras, 26500 Patras, Greece

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

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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