Abstract
Discrete Chebyshev-type inequalities are established for sequences of modified Bessel functions of the first and second kind, recognizing that the sums involved are actually Neumann series of modified Bessel functions I ν and K ν . Moreover, new closed integral expression formulae are established for the Neumann series of second type, which occur in the discrete Chebyshev inequalities.
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Notes
- 1.
It is worth to mention here that the above procedure for modified Bessel functions is similar of the method for Bessel functions applied by Wilkins [27]. See also Andrews et al. [1] for more details. More precisely, Wilkins proved that the Hankel functions \(\big{(H_{\nu }^{(1)}\big)}^{2}\) and \(\big{(H_{\nu }^{(2)}\big)}^{2}\), as well as \(J_{\nu }^{2} + Y _{\nu }^{2}\), where J ν and Y ν stand for the Bessel functions of the first and second kind, are particular solutions of the third-order homogeneous differential equation [1, p. 225]
$$\displaystyle{{x}^{2}y^{\prime\prime\prime}(x) + 3xy^{\prime\prime}(x) + (1 + 4{x}^{2} - {4\nu }^{2})y^{\prime}(x) + 4xy(x) = 0.}$$The above result was used to prove the celebrated Nicholson formula [1, p. 224]
$$\displaystyle{J_{\nu }^{2}(x) + Y _{\nu }^{2}(x) ={ \frac{8} {\pi }^{2}}\int _{0}^{\infty }K_{ 0}(2x\sinh t)\cosh (2\nu t)dt,}$$which generalizes the trigonometric identity \({\sin }^{2}x {+\cos }^{2}x = 1\).
References
Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Baricz, Á.: On a product of modified Bessel functions. Proc. Amer. Math. Soc. 137(1), 189–193 (2009)
Baricz, Á.: Bounds for modified Bessel functions of the first and second kinds. Proc. Edinb. Math. Soc. 53(3), 575–599 (2010)
Baricz, Á., Jankov, D., Pogány, T.K.: Integral representations for Neumann-type series of Bessel functions I ν , Y ν and K ν . Proc. Amer. Math. Soc. 140(2), 951–960 (2012)
Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)
Baricz, Á., Pogány, T.K.: Turán determinants of Bessel functions. Forum Math. (2011 in press)
Baricz, Á., Ponnusamy, S.: On Turán type inequalities for modified Bessel functions. Proc. Amer. Math. Soc. 141(2), 523–532 (2013)
Cahen, E.: Sur la fonction ζ(s) de Riemann et sur des fontions analogues. Ann. Sci. l’École Norm. Sup. Sér. 11, 75–164 (1894).
Cochran, J.A.: The monotonicity of modified Bessel functions with respect to their order. J. Math. Phys. 46, 220–222 (1967)
Graham, R.L.: Application of the FKG Inequality and its Relatives, Mathematical Programming: The State of the Art (Bonn, 1982), pp. 115–131. Springer, Berlin (1983)
Grandison, S., Penfold, R., Vanden-Broeck, J.M.: A rapid boundary integral equation technique for protein electrostatics. J. Comput. Phys. 224, 663–680 (2007)
Hasan, A.A.: Electrogravitational stability of oscillating streaming fluid cylinder. Phys. B. 406, 234–240 (2011)
van Heijster, P., Doelman, A., Kaper, T.J.: Pulse dynamics in a three-component system: stability and bifurcations. Phys. D. Nonlinear Phenomena 237(24), 3335–3368 (2008)
van Heijster, P., Doelman, A., Kaper, T.J., Promislow, K.: Front interactions in a three-component system. SIAM J. Appl. Dyn. Syst. 9, 292–332 (2010)
van Heijster, P., Sandstede, B.: Planar radial spots in a three-component FitzHugh-Nagumo system. J. Nonlinear Sci. 21, 705–745 (2011)
Jones, A.L.: An extension of an inequality involving modified Bessel functions. J. Math. Phys. 47, 220–221 (1968)
Klimek, S., McBride, M.: Global boundary conditions for a Dirac operator on the solid torus. J. Math. Phys. 52, Article 063518, 14 pp (2011)
Laforgia, A.: Bounds for modified Bessel functions. J. Computat. Appl. Math. 34(4), 263–267 (1991)
Penfold, R., Vanden-Broeck, J.M., Grandison, S.: Monotonicity of some modified Bessel function products. Integral Transforms Spec. Funct. 18, 139–144 (2007)
Perron, O.: Zur Theorie der Dirichletschen Reihen. J. Reine Angew. Math. 134, 95–143 (1908)
Phillips, R.S., Malin, H.: Bessel function approximations. Amer. J. Math. 72, 407–418 (1950)
Pogány, T.K., Süli, E.: Integral representation for Neumann series of Bessel functions. Proc. Amer. Math. Soc. 137(7), 2363–2368 (2009)
Radwan, A.E., Dimian, M.F., Hadhoda, M.K.: Magnetogravitational stability of a bounded gas-core fluid jet. Appl. Energy 83, 1265–1273 (2006)
Radwan, A.E., Hasan, A.A.: Magneto hydrodynamic stability of self-gravitational fluid cylinder. Appl. Math. Modell. 33, 2121–2131 (2009)
Reudink, D.O.: On the signs of the ν-derivatives of the modified Bessel functions I ν (x) and K ν (x). J. Res. Nat. Bur. Standards B72, 279–280 (1968)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)
Wilkins, J.E.: Nicholson’s integral for \(J_{n}^{2}(z) + Y _{n}^{2}(z)\). Bull. Amer. Math. Soc. 54, 232–234 (1948)
Acknowledgements
The authors are grateful to Christoph Koutschan who provided expert help in deriving the differential equation (17).
The research of Á. Baricz was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2012-3-0190.
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Baricz, Á., Pogány, T.K. (2014). Properties of the Product of Modified Bessel Functions. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_31
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