Properties of the Product of Modified Bessel Functions

  • Árpád Baricz
  • Tibor K. Pogány


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.



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.


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Baricz, Á.: On a product of modified Bessel functions. Proc. Amer. Math. Soc. 137(1), 189–193 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baricz, Á.: Bounds for modified Bessel functions of the first and second kinds. Proc. Edinb. Math. Soc. 53(3), 575–599 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    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)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Baricz, Á., Pogány, T.K.: Turán determinants of Bessel functions. Forum Math. (2011 in press)Google Scholar
  7. 7.
    Baricz, Á., Ponnusamy, S.: On Turán type inequalities for modified Bessel functions. Proc. Amer. Math. Soc. 141(2), 523–532 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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).MATHMathSciNetGoogle Scholar
  9. 9.
    Cochran, J.A.: The monotonicity of modified Bessel functions with respect to their order. J. Math. Phys. 46, 220–222 (1967)MATHMathSciNetGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Grandison, S., Penfold, R., Vanden-Broeck, J.M.: A rapid boundary integral equation technique for protein electrostatics. J. Comput. Phys. 224, 663–680 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hasan, A.A.: Electrogravitational stability of oscillating streaming fluid cylinder. Phys. B. 406, 234–240 (2011)CrossRefGoogle Scholar
  13. 13.
    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)CrossRefMATHGoogle Scholar
  14. 14.
    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)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    van Heijster, P., Sandstede, B.: Planar radial spots in a three-component FitzHugh-Nagumo system. J. Nonlinear Sci. 21, 705–745 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Jones, A.L.: An extension of an inequality involving modified Bessel functions. J. Math. Phys. 47, 220–221 (1968)MATHGoogle Scholar
  17. 17.
    Klimek, S., McBride, M.: Global boundary conditions for a Dirac operator on the solid torus. J. Math. Phys. 52, Article 063518, 14 pp (2011)Google Scholar
  18. 18.
    Laforgia, A.: Bounds for modified Bessel functions. J. Computat. Appl. Math. 34(4), 263–267 (1991)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Penfold, R., Vanden-Broeck, J.M., Grandison, S.: Monotonicity of some modified Bessel function products. Integral Transforms Spec. Funct. 18, 139–144 (2007)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Perron, O.: Zur Theorie der Dirichletschen Reihen. J. Reine Angew. Math. 134, 95–143 (1908)MATHGoogle Scholar
  21. 21.
    Phillips, R.S., Malin, H.: Bessel function approximations. Amer. J. Math. 72, 407–418 (1950)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Pogány, T.K., Süli, E.: Integral representation for Neumann series of Bessel functions. Proc. Amer. Math. Soc. 137(7), 2363–2368 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Radwan, A.E., Dimian, M.F., Hadhoda, M.K.: Magnetogravitational stability of a bounded gas-core fluid jet. Appl. Energy 83, 1265–1273 (2006)CrossRefGoogle Scholar
  24. 24.
    Radwan, A.E., Hasan, A.A.: Magneto hydrodynamic stability of self-gravitational fluid cylinder. Appl. Math. Modell. 33, 2121–2131 (2009)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    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)MathSciNetGoogle Scholar
  26. 26.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)MATHGoogle Scholar
  27. 27.
    Wilkins, J.E.: Nicholson’s integral for \(J_{n}^{2}(z) + Y _{n}^{2}(z)\). Bull. Amer. Math. Soc. 54, 232–234 (1948)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of EconomicsBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Faculty of Maritime StudiesUniversity of RijekaRijekaCroatia

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