Integral Representations of Functional Series with Members Containing Jacobi Polynomials

MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10In this article we establish a double definite integral representation, and two other indefinite integral expressions for a functional series and its derivative with members containing Jacobi polynomials

NEW DOCTORAL DEGREES Integral expressions for series of functions of hypergeometric and Bessel types

This thesis presents some new results on integral expressions for series of functions of hypergeometric and Bessel types. Also there are derived two--sided inequalities of some hypergeometric functions, which are related with their integral representations. In the first part of the thesis are defined some special functions, mathematical methods, and results which we use in prooving our own. Some of them are Gamma function, Gauss hypergeometric function ${}_2F_1$ and generalized hypergeometric function ${}_pF_q$. There are also Fox-Wright generalized hypergeometric function ${}_pPsi_q$ and the Struve function $textbf{H}_nu(z)$.\ Bessel differential equation is also described, and that is one of the crucial mathematical tools that we use.\ Mathieu $(boldsymbol a, boldsymbol lambda)$-- and Dirichlet series are defined too, because they are useful for deriving most of integral representations. In that purpose, we also use condensed form of Euler--Maclaurin summation formula and fractional analysis, which are described in the introduction. In the middle part of the thesis, i.e. in Chapter 3, 4 and 5 we work on integral representations of functional series with members containing Bessel functions of the first kind, which are divided into three types: Neumann series, which are discussed in Chapter 3, Kapteyn series, which are described in Chapter 4, and Schl"omilch series, which are observed in Chapter 5. In the last chapter of this thesis, we obtain a functional series of hypergeometric types. There, we also derive an integral representations of hypergeometric functions, such as extended general Hurwitz--Lerch Zeta function and extended Hurwitz--Lerch Zeta function, and also the two-sided inequalities for the mentioned functions. \At the end of this chapter, new incomplete generalized Hurwitz--Lerch Zeta functions and incomplete generalized Gamma functions are defined, and we also investigate their important properties

Egyptian fractions

U ovome radu su prezentirane osnovne odrednice matematike starih Egipćana, s posebnim naglaskom na vrstu razlomaka koje su oni poznavali, te ih stoga nazivamo egipatski razlomci ili jedinični razlomci. Rad sadrži i slike na kojima su predstavljeni posebni simboli koje su za brojeve dekadskog sustava i jedinične razlomke koristili stari Egipćani.In this paper we present the basic principles of mathematics of the ancient Egyptians, with special emphasis on the type of fractions that they used, so they called Egyptian fractions, or unit fractions. The paper also includes images which present the special symbols which were used by Egyptians, as the symbols for decimal system and unit fractions

Bounds for confluent Horn function Φ2 deduced by McKay Iν Bessel law

The main aim of this article is to derive by probabilistic method new functional and uniform bounds for Horn confluent hypergeometric Φ2 of two variables and the incomplete Lipschitz–Hankel integral, among others. The main mathematical tools are the representation theorems for the McKay Iν Bessel probability distribution\u27s CDF and certain known and less known properties of cumulative distribution functions

Mjera s predznakom: može li rezultat mjerenja biti negativan broj?

U ovom članku upoznat ćemo se s pojmom mjere s predznakom i njezinim rastavom na razliku dviju mjera. Ovakav rastav omogućuju nam dva teorema koja su u literaturi poznata kao Hahnova i Jordanova dekompozicija mjere s predznakom. članak također sadrži nekoliko ilustrativnih primjera koji konkretiziraju navedene pojmove