Slobodan B. Tričković

On Euler-like methods for the simultaneous approximation of polynomial zeros

In this paper we consider some iterative methods of higher order for the simultaneous determination of polynomial zeros. The proposed methods are based on Euler’s third order method for finding a zero of a given function and involve Weierstrass’ correction in the case of simple zeros. We prove that the presented methods have the order of convergence equal to four or more. Based on a fixed-point relation of Euler’s type, two inclusion methods are derived. Combining the proposed methods in floating-point arithmetic and complex interval arithmetic, an efficient hybrid method with automatic error bounds is constructed. Computational aspect and the implementation of the presented algorithms on parallel computers are given.

On the Orthogonality of Classical Orthogonal Polynomials

We consider the orthogonality of rational functions W n ( s ) as the Laplace transform images of a set of orthoexponential functions, obtained from the Jacobi polynomials, and as the Laplace transform images of the Laguerre polynomials. We prove that the orthogonality of the Jacobi and the Laguerre polynomials is induced by the orthogonality of the functions W n ( s ). Thus we have shown that the orthogonality relations of the Jacobi and Laguerre polynomials are equivalent to the orthogonality of rational functions which are essentially the images of the classical orthogonal polynomials under the Laplace transform.

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.

On zero-finding methods of the fourth order

Using the iteration formulas of the third order for solving the single equation f(z) = 0 and a procedure for the acceleration of convergence, three new methods of the fourth order are derived. The comparison with other methods is given.

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.

A new approach to the orthogonality of the Laguerre and Hermite polynomials

This article draws on results from [Tričković, S.B. and Stanković, M.S., 2003, On the orthogonality of classical orthogonal polynomials. Integral Transforms and Special Functions, 14 (3), 271–280.], where we considered the orthogonality of rational functions W n (s) which are obtained as the images of the classical orthogonal polynomials under the Laplace transform. We proved in [Tričković, S.B. and Stanković, M.S., 2003, On the orthogonality of classical orthogonal polynomials. International Transaction of Specific Function, 14 (3), 271–280.] that the orthogonality relations of the Jacobi polynomials and the standard Laguerre polynomials L n (x) are induced by and are equivalent to the orthogonality of rational functions W n (s). In this article, we continue in the same manner by considering the generalized Laguerre polynomials and Hermite polynomials H n (x). In the last section, we analyze the zeros distribution of the Laplace transform images of the Legendre, Chebyshev, Laguerre and Hermite polynomials.

On A Method for the Construction of Some Q -Orthogonal Polynomials

This paper fully relies on the results of our paper [12] where we considered the orthogonality of rational functions

On trigonometric series over integrals involving Bessel or Struve functions

W_n(s)

Summation of Series over Bourget Functions

as the Laplace transform images of a set of orthoexponential functions, obtained from Jacobi polynomials, and as the Laplace transform images of Laguerre polynomials. In Ref. [12] we proved that the orthogonality of Jacobi and Laguerre polynomials is induced by the orthogonality of the functions