Skip to main content
Log in

On coefficients of Kapteyn-type series

Mathematica Slovaca

Cite this article


Quite recently Jankov and Pogány [JANKOV, D.—POGÁNY, T. K.: Integral representation of Schlömilch series, J. Classical Anal. 1 (2012) 75–84] derived a double integral representation of the Kapteyn-type series of Bessel functions. Here we completely describe the class of functions Λ = {α}, which generate the mentioned integral representation in the sense that the restrictions \(\alpha |_\mathbb{N} = (\alpha _n )_{n \in \mathbb{N}} \) is the sequence of coefficients of the input Kapteyn-type series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions


  1. ANDREWS, G. E.— ASKEY, R.— ROY, R.: Special Functions. EncyclopediaMath. Appl. 71, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  2. BARICZ, Á.— JANKOV, D.— POGÁNY, T. K.: Integral representation of first kind Kapteyn series, J. Math. Phys. 52 (2011), Article ID 043518.

  3. CITRIN, D. S.: Optical analogue for phase-sensitive measurements in quantum-transport experiments, Phys. Rev. B 60 (1999), 5659–5663.

    Article  Google Scholar 

  4. DOMINICI, D.: A new Kapteyn series, Integral Transforms Spec. Funct. 18 (2007), 409–418.

    Article  MATH  MathSciNet  Google Scholar 

  5. DOMINICI, D.: An application of Kapteyn series to a problem from queueing theory, Proc. Appl. Math. Mech. 7 (2007), 2050005–2050006.

    Article  Google Scholar 

  6. DOMINICI, D.: On Taylor series and Kapteyn series of the first and second type, J. Comput. Appl. Math. 236 (2011), 39–48.

    Article  MATH  MathSciNet  Google Scholar 

  7. EISINBERG, A.— FEDELE, G.— FERRISE, A.— FRASCINO, D.: On an integral representation of a class of Kapteyn (Fourier-Bessel) series: Kepler’s equation, radiation problems and Meissel’s expansion, Appl. Math. Lett. 23 (2010), 1331–1335.

    Article  MATH  MathSciNet  Google Scholar 

  8. JANKOV, D.— POGÁNY, T. K.: Integral representation of Schlömilch series, J. Classical Anal. 1 (2012), 75–84.

    Google Scholar 

  9. JANKOV, D.— POGÁNY, T. K.— SÜLI, E.: On the coefficients of Neumann series of Bessel functions, J. Math. Anal. Appl. 380 (2011), 628–631.

    Article  MATH  MathSciNet  Google Scholar 

  10. KAPTEYN, W.: Recherches sur les functions de Fourier-Bessel, Ann. Sci. Éc. Norm. Supér. (4) 10 (1893), 91–120.

    MATH  MathSciNet  Google Scholar 

  11. KAPTEYN, W.: On an expansion of an arbitrary function in a series of Bessel functions, Messenger of Math. 35 (1906), 122–125.

    Google Scholar 

  12. LANDAU, L.: Monotonicity and bounds on Bessel functions. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory, Berkeley, California (2000), 147–154 (Electronic); Electron. J. Differ. Equ. Conf. 4, Southwest Texas State University, San Marcos, TX, 2000.

    Google Scholar 

  13. LERCHE, I.— SCHLICKEISER, R.— TAUTZ, R. C.: Comment on a new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions, Physics of Plasmas 15 (2008), Article ID 024701.

  14. LERCHE, I.— TAUTZ, R. C.: A note on summation of Kapteyn series in astrophysical problems, Astrophys. J. 665 (2007), 1288–1291.

    Article  Google Scholar 

  15. LERCHE, I.— TAUTZ, R. C.: Kapteyn series arising in radiation problems, J. Phys. A 41 (2008), Article ID 035202.

    Google Scholar 

  16. LERCHE, I.— TAUTZ, R. C.— CITRIN, D. S.: Terahertz-sideband spectra involving Kapteyn series, J. Phys. A 42 (2009), Article ID 365206.

  17. MARSHALL, T. A.: On the sums of a family of Kapteyn series, Z. Angew. Math. Phys. 30 (1979), 1011–1016.

    Article  MATH  MathSciNet  Google Scholar 

  18. NIELSEN, N.: Recherches sur les séries de fonctions cylindriques dues á C. Neumann et W. Kapteyn, Ann. sci. de l’École Norm. Sup. 18 (1901), 39–75.

    MATH  Google Scholar 

  19. PLATZMAN, G. W.: An exact integral of complete spectral equations for unsteady onedimensional flow, Tellus 4 (1964), 422–431.

    Article  MathSciNet  Google Scholar 

  20. SCHOTT, G. A.: Electromagnetic Radiation and the Mechanical Reactions Arising From It, Being an Adams Prize Essay in the University of Cambridge, Cambridge University Press, Cambridge, 1912.

    Google Scholar 

  21. SHALCHI, A.— SCHLICKEISER, R.: Cosmic ray transport in anisotropic magnetohydrodynamic turbulence III. Mixed magnetosonic and Alfvènic turbulence, Astronom. Astrophys. 420 (2004), 799–808.

    Article  Google Scholar 

  22. TAUTZ, R. C.— LERCHE, I.: A review of procedures for summing Kapteyn series in mathematical physics, Adv. Math. Phys. 2009 (2009), Article ID 425164.

  23. THOMSON, J. J.: The magnetic properties of systems of corpuscles describing circular orbits, Philos. Mag. 6 (1903), 673–693.

    Article  MATH  Google Scholar 

  24. WATSON, G. N.: A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dragana Jankov.

Additional information

Communicated by Ján Borsík

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jankov, D., Pogány, T. On coefficients of Kapteyn-type series. Math. Slovaca 64, 403–410 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

2010 Mathematics Subject Classification