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Results in Mathematics

, 75:12 | Cite as

Second Type Neumann Series of Generalized Nicholson Function

  • Dragana Jankov MaširevićEmail author
  • Tibor K. Pogány
Article
  • 25 Downloads

Abstract

The second type Neumann series are considered whose building blocks are generalized Nicholson’s functions \(B_\nu ^p(x) :=J_\nu ^p(x)+ Y_\nu ^p(x)\), being \(J_\nu , Y_\nu \) Bessel functions of the first and second kind of order \(\nu \), \(p \ge 2\) integer. Closed form definite integral expressions are obtained for such series with the aid of the associated Dirichlet series’ Cahen’s Laplace integral form.

Keywords

Bessel functions of the first and second kind modified Bessel function of the second kind \(K_\nu \) Nicholson function Neumann series of Bessel functions Cahen’s Laplace integral formula 

Mathematics Subject Classification

Primary 33C10 33E20 40C10 Secondary 33E30 40H05 

Notes

Acknowledgements

The authors are indebted to both unknown referees for several constructive comments which mainly improve the exposition’s relevance and completeness and finally encompass the article. T.K. Pogány acknowledges the support given by the NAWA project PROM PPI/PRO/2018/1/00008 and thanks to the Department of Mathematical Physics, The Henryk Niewodniczański Institute of Nuclear Physics of Polish Academy of Sciences, Kraków, Poland for the warm hospitality and the excellent working atmosphere during his stay there during February 2019.

References

  1. 1.
    Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists: A Comprehensive Guide, 7th edn. Elsevier, Oxford (2013)zbMATHGoogle Scholar
  2. 2.
    Baricz, Á., Jankov, D., Pogány, T.K.: Integral representations for Neumann-type series of Bessel functions \(I_{\nu },\) \(Y_{\nu }\) and \(K_{\nu }\). Proc. Amer. Math. Soc. 140(2), 951–960 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baricz, Á., Jankov Maširević, D., Pogány, T.K.: Series of Bessel and Kummer-Type Functions. Lecture Notes in Mathematics, vol. 2207. Springer, Cham (2017)zbMATHCrossRefGoogle Scholar
  5. 5.
    Baricz, Á., Pogány, T.K.: Properties of the product of modified Bessel functions. In: Milovanović, G.V., Rassias, M.T. (eds.) Analytic Number Theory, Approximation theory, and Special Functions, pp. 809–820. Springer, New York (2014). In Honor of Hari M. SrivastavazbMATHCrossRefGoogle Scholar
  6. 6.
    Baricz, Á., Pogány, T.K., Ponnusamy, S., Rudas, I.: Bounds for Jaeger integrals. J. Math. Chem. 53(5), 1257–1273 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Beranek, L.L., Mellow, T.J.: Acoustics: Sound Fields and Transducers. In: Chapter 12 :Radiation and scattering of sound by the boundary value method, pp. 487–533. Academic Press, Cambridge (2012)CrossRefGoogle Scholar
  8. 8.
    Cahen, E.: Sur la fonction \(\zeta (s)\) de Riemann et sur des fonctions analogues. Ann. Sci. l’École Norm. Sup. (Sér. 3) 11, 75–164 (1894)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Carlslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Oxford University Press, Oxford (1959)Google Scholar
  10. 10.
    Cochran, J.A.: Three-dimensional temperature response to impulsive input outside a spherical reservoir. SIAM J. Math. Anal. 18(5), 283–290 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dunster, T.M.: On the logarithmic derivative of Nicholson’s integral. Anal. Appl. 7(1), 73–86 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dunster, T.M., Yedlin, M., Lam, K.: Resonance and the late coefficients in the scattered field of a dielectric circular cylinder. Anal. Appl. 4(4), 311–333 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Durand, L.: Product formulas and Nicholson-type integrals for Jacobi functions. I: Summary of results. SIAM J. Math. Anal. 9(1), 76–86 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Freitas, P.: Sharp bounds for the modulus and phase of Hankel functions with applications to Jaeger integrals. Math. Comp. 87(309), 289–308 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, (2000)Google Scholar
  16. 16.
    Hardy, G.H., Riesz, M.: The General Theory of Dirichlet’s Series. University Press, Cambridge (1915)zbMATHGoogle Scholar
  17. 17.
    Harrington, R.F.: Time–Harmonic Electromagnetic Fields. IEEE Press Series on Electromagnetic Wave Theory. Wiley, New York (2001)CrossRefGoogle Scholar
  18. 18.
    Homicz, G.F., Lordi, J.A.: A note on the radiative directivity patterns of duct acoustic modes. J. Sound Vib. 41(3), 283–290 (1975)CrossRefGoogle Scholar
  19. 19.
    Horvat, M., Prosen, T.: The bends on a quantum waveguide and cross-products of Bessel functions. J. Phys. A: Math. Theor. 40, 6349–6379 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hrycak, T., Schmutzhard, S.: A Nicholson-type integral for the cross-product of the Bessel functions. J. Math. Anal. Appl. 436(1), 168–178 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hsu, Y.K.: A brief review of supercavitating hydrofoils. J. Hydronautics 2(4), 192–197 (1968)CrossRefGoogle Scholar
  22. 22.
    Jaeger, J.C.: Conduction of heat in regions bounded by planes and cylinders. Bull. Am. Math. Soc. 47(10), 734–741 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Karamata, J.: Theory and Applications of Stieltjes integral. Srpska Akademija Nauka, Posebna izdanja CLIV, Matematički institut, Knjiga I, Beograd (1949). (in Serbian)Google Scholar
  24. 24.
    Knopp, K.: Theorie und Anwendungen der unendlichen Reihen, Vierte edn. Springer, Berlin (1947)zbMATHCrossRefGoogle Scholar
  25. 25.
    Korenev, B.G.: Bessel Functions and their Applications.Translated from the Russian by Pankratiev, E.V. Analytical Methods and Special Functions, Taylor & Francis Ltd, London (2002)Google Scholar
  26. 26.
    Krieger, L., Roth, M., von der Lühe, O.: Estimating the solar meridional circulation by normal mode decomposition. Astron. Nachr. 328(3/4), 252–256 (2007)zbMATHCrossRefGoogle Scholar
  27. 27.
    Martinec, Z.: Thomson-Haskell matrix method for free spheroidal elastic oscillations. Geophys. J. Int. 98, 195–199 (1989)CrossRefGoogle Scholar
  28. 28.
    Muskhelishvili, N.I.: Singular Integral Equations.Boundary Problems of Function Theory and Their Application to Mathematical Physics, Translation from Russian by J.R.M. Radok. P. Noordhoff N. V., Groningen, Holland (1953)zbMATHGoogle Scholar
  29. 29.
    Nicholson, J.W.: The asymptotic expansions of Bessel functions. Phil. Mag. 19(6), 228–249 (1910)zbMATHCrossRefGoogle Scholar
  30. 30.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. NIST and Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  31. 31.
    Phillips, W.R.C., Mahon, P.J.: On approximations to a class of Jaeger integrals. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2136), 3570–3589 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Perron, O.: Zur Theorie der Dirichletschen Reihen. J. Reine Angew. Math. 134, 95–143 (1908)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pogány, T.K.: Integral representation of a series which includes the Mathieu a-series. J. Math. Anal. Appl. 296(1), 309–313 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Pogány, T.K.: Multiple Euler-McLaurin summation formula. Mat. Bilt. 29, 37–40 (2005)zbMATHGoogle Scholar
  35. 35.
    Pogány, T.K.: Integral representation of Mathieu \((\varvec {a}, \varvec {\lambda })\)-series. Integral Transforms Spec. Funct. 16(5), 685–689 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Pogány, T.K., Süli, E.: Integral representation for Neumann series of Bessel functions. Proc. Amer. Math. Soc. 137(7), 2363–2368 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Pogány, T.K., Srivastava, H.M., Tomovski, Ž.: Some families of Mathieu \(\varvec {a}\)-series and alternating Mathieu \(\varvec {a}\)-series. Appl. Math. Comput. 173(1), 69–108 (2006)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integrals and Series. Direct Laplace Transforms. Gordon and Breach Science Publishers, New York,Reading, Paris,Montreux, Tokyo, Melbourne (1992)zbMATHGoogle Scholar
  39. 39.
    Simon, B.: Harmonic Analysis. A Comprehensive Course in Analysis Part 3. American Mathematical Society, Providence, Rhode Island (2015)zbMATHGoogle Scholar
  40. 40.
    Smith, L.P.: Heat flow in an infinite solid bounded internally by a cylinder. J. Appl. Phys. 8, 441–448 (1937)zbMATHCrossRefGoogle Scholar
  41. 41.
    Tsao, C.Y.H., Payne, D.N., Gambling, W.A.: Modal characteristics of three-layered optical fiber waveguides: a modified approach. J. Opt. Soc. Am. A 6(4), 555–563 (1989)CrossRefGoogle Scholar
  42. 42.
    Vilenkin, N.J.: Special functions and the Theory of Group representations. Translated from the Russian by Singh, V. N. Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island (1968)zbMATHGoogle Scholar
  43. 43.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dragana Jankov Maširević
    • 1
    Email author
  • Tibor K. Pogány
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of OsijekOsijekCroatia
  2. 2.Faculty of Maritime StudiesUniversity of RijekaRijekaCroatia
  3. 3.Institute of Applied MathematicsÓbuda UniversityBudapestHungary

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