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Summation of Series over Bourget Functions

  • Mirjana V. Vidanović (a1), Slobodan B. Tričković (a2) and Miomir S. Stanković (a1)
Abstract

In this paper we derive formulas for summation of series involving J. Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H. M. Srivastava. These series are expressed in terms of the Riemann ζ function and Dirichlet functions η, λ, β, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

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References
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[1] Abramowitz, M. and Stegun, A., eds., Handbook ofMathematical Functions, with Formulas, Graphs and Mathematical Tables. Dover, New York, 1972.
[2] Bourget, J., Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste. J. de Math.(2) 6(1861), 3354.
[3] Gorowara, K. K., On Bourget's function Jn,k(z). Ganita 22(1971), no. 1, 2126.
[4] Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series and Products. Sixth edition. Academic Press, San Diego, CA, 2000.
[5] Prudnikov, A. P., Brychkov, Y. A., and Marichev, O. I., Integrals and Series, Volume 1, Elementary Functions. Gordon and Breach Science Publishers, New York, 1986.
[6] Srivastava, H. M., A note on functions analogous to Bourget's function. Ganita 19(1968), no. 2, 4548.
[7] Stanković, M. S., Vidanović, M. V., and Tričković, S. B., On the summation of series involving Bessel or Struve functions. J. Math. Anal. Appl. 247(2000), no. 1, 1526.
[8] Tričković, S. B., Vidanović, M. V., and Stanković, M. S., On the summation of series in terms of Bessel functions. Z. Anal. Anwend. 25(2006), no. 3, 393406.
[9] Tričković, S. B., Vidanović, M. V., and Stanković, M. S. On the summation of trigonometric series. Integral Transforms Spec. Funct. 19(2008), no. 6, 441452.
[10] Watson, G. N., A Treatise on the Theory of Bessel Functions. Second edition. Cambridge University Press, Cambridge, 1944.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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