Mirjana V. Vidanović

On the Summation of Series in Terms of Bessel Functions

In this article we deal with summation formulas for the series ∑ ∞ n=1 Jμ(nx) nν , referring partly to some results from our paper in J. Math. Anal. Appl. 247 (2000) 15 – 26. We show how these formulas arise from different representations of Bessel functions. In other words, we first apply Poisson’s or Bessel’s integral, then in the sequel we define a function by means of the power series representation of Bessel functions and make use of Poisson’s formula. Also, closed form cases as well as those when it is necessary to take the limit have been thoroughly analyzed.

On the summation of trigonometric series

We deal with the series and express it as a power series in terms of Riemanns ζ or Catalans β function or Dirichlet functions η and λ. Also, closed form cases as well as those when it is necessary to take limit have been thoroughly analyzed. Some applications such as convergence acceleration are considered too.

Series over the product of bessel and trigonometric functions

This paper is a contribution to the mathematics of Bessel series which is used extensively in engineering. The sums of series (1) and (2) are found by considering series involving the product of two trigonometric functions. The obtained sums are in terms of Riemann zeta and related functions.

Series involving the product of a trigonometric integral and a trigonometric function

This paper is concerned with the summation of series (1). To find the sum of the series (1) we first derive formulas for the summation of series whose general term contains a product of two trigonometric functions. These series are expressed in terms of Riemanns zeta, Catalans beta function or Dirichlet functions eta and lambda, and in certain cases, thoroughly investigated here, they can be brought in closed form, meaning that the infinite series are represented by finite sums.

On trigonometric series over integrals involving Bessel or Struve functions

To find formulas for the summation of trigonometric series over integrals involving Bessel or Struve functions, we rely on trigonometric series involving Bessel or Struve functions, which are in turn obtained by using summation formulas for series over the product of two trigonometric functions. All these sums are expressed either as power series in terms of Riemann’s ζ or Catalan’s β function or Dirichlet functions η and λ, or, in certain cases, they are brought in so called closed form, which means that the infinite series are represented by finite sums. Important limiting values cases are considered too.

Summation of Series over Bourget Functions

In this paper we derive formulas for summationof 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. (M.V. M.S.) Department of Mathematics, Faculty of Environmental Engineering, University of Nǐs, 18000 Nǐs, Serbia e-mail: mvv@znrfak.ni.ac.yu mstan@znrfak.ni.ac.yu (S.T.) Department of Mathematics, Faculty of Civil Engineering, University of Nǐs, 18000 Nǐs, Serbia e-mail: sbt@gaf.ni.ac.yu Received by the editors May 23, 2006; revised November 26, 2006. This work is supported by the Ministry of Science of Serbia. AMS subject classification: Primary: 33C10; secondary: 11M06, 65B10.

On the summation of series over a product of Bessel functions

In this article, we deal with summation formulas for the series (3), referring partly to some results from our article [Stanković, M.S., Vidanović, M.V. and Tric˘ković, S.B., 2000, On the summation of series in involving Bessel or Struve functions. Journal of Mathematics and Analytical Applications, 247, 15–26]. We show how these formulas arise from different representations of Bessel functions. In other words, we first apply Gegenbauers integral, then in the sequel we define a function by means of the power series representation of Bessel functions and make use of Poissons formula. Also, closed-form cases as well as those when it is necessary to take limit have been thoroughly analyzed.

Summation of Some Trigonometric and Schlömilch Series

In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylovs method and our formula (7).

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

The sum of the series Sα = Sα s, a, b, f(y), g(x) = ∞ X n=1 (s)n−1f (an− b)y g (an− b)x (an− b)α involving the product of two trigonometric functions is obtained using the sum of the series ∞ X n=1 (s)(n−1)f((an− b)x) (an− b)α = cπ 2Γ(α)f( 2 ) xα−1 + ∞ X i=0 (−1) F (α− 2i− δ) (2i + δ)! x 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.

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.