JournalszaaVol. 20, No. 1pp. 235–246

Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions

  • Miomir S. Stankovic

    University of Nis, Serbia
  • Mirjana V. Vidanovic

    University of Nis, Serbia
  • Slobodan B. Trickovic

    University of Nis, Serbia
Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions cover
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Abstract

The sum of the series

Sα=Sαs,a,b,f(y),g(x)=n=1(s)n1f(anb)yg(anb)x(anb)αS_{\alpha} = S_{\alpha} s,a,b,f(y),g(x) = \sum^\infty_{n=1} \frac {(s)^{n–1}f (an – b)y g (an–b)x}{(an–b)^{\alpha}}

involving the product of two trigonometric functions is obtained using the sum of the series

n=1(s)(n1)f((anb)x)(anb)α=cπ2Γ(α)f(πα2)xα1+i=0(1)iF(α2iδ)(2i+δ)!x2i+δ\sum^\infty_{n=1} \frac {(s)^{(n–1)} f((an–b)x)}{(an–b)^\alpha} = \frac {c\pi}{2\Gamma (\alpha) f (\frac {\pi \alpha}{2})} x^{\alpha–1} + \sum^\infty_{i=0} (–1)^i \frac {F(\alpha – 2i – \delta)}{(2i + \delta)!} x^{2i+\delta}

whose terms involve one trigonometric function. The first series is represented as series in terms of the Riemann zeta and related functions, which has a closed form in certain cases. Some applications of these results to the summation of series containing Bessel functions are given. The obtained results also include as special cases formulas in some known books. We further show how to make use of these results to obtain closed form solutions of some boundary value problems in mathematical physics.

Cite this article

Miomir S. Stankovic, Mirjana V. Vidanovic, Slobodan B. Trickovic, Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions. Z. Anal. Anwend. 20 (2001), no. 1, pp. 235–246

DOI 10.4171/ZAA/1014