engleski

Integral expressions for series of functions of hypergeometric and Bessel types

In the 18th century, it became clear that the existing elementary functions are not sufficient to describe a number of unsolved problems in various branches of mathematics and physics. Functions that describe the results generally presented in the form of infinite series, integrals, or as solutions of differential equations are collectively called special functions. The classical book devoted to Bessel functions is the famous monograph by G.N. Watson, A Treatise on the Theory of Bessel Functions, 1922. The first main goal of the thesis is to establish integral representation formulae for the so--called Neumann series of the Bessel functions of the first and second kind and the modified Bessel functions of the first and second kind following traces of the article by Pogany and Suli. The same will be realized for Kapteyn and Schloemilch series. This will be done by associated Dirichlet series, Euler-Maclaurin summation formula from one, and using Bessel differential equation and fractional differinte-gral representation of thier solutions. The second capital set of results concerns the various extensions, unifications and generali-zations of the so- called Hurwitz-Lerch Zeta function. Among others new closed form integral representations and sharp bilateral bounding inequalities have been established. Part of the achieved results are published in 7 consecutive papers written in couathorship with A. Baricz, E. Suli, H.M. Srivastava, R.K. Saxena and T. Pogany. Few further titles under consideration too.

Bessel functions of the first and second kind ; Modified Bessel functions of the first and second kind ; Neumann series of Besssel functions ; Kapteyn series ; Schloemilch series ; Integral represntatioon ; Dirichlet series ; Hurwitz-Lerch Zeta function ; Bessel differential equation ; Two-sided bounding inequalities

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