Dragana Jankov

Turán type inequalities for Krätzel functions

Abstract Complete monotonicity, Laguerre and Turan type inequalities are established for the so-called Kratzel function Z ρ ν , defined by Z ρ ν ( u ) = ∫ 0 ∞ t ν − 1 e − t ρ − u t d t , where u > 0 and ρ , ν ∈ R . Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Kratzel function.

Integral representations for Neumann-type series of Bessel functions Iv, Yv and Kv

Recently Pogány and Süli [Proc. derived a closed-form integral expression for Neumann series of Bessel functions of the first kind J ν. In this paper our aim is to establish analogous integral representations for the Neumann-type series of modified Bessel functions of the first kind I ν and for Bessel functions of the second kind Y ν , K ν , and to give links for the same question for the Hankel functions H (1) ν , H (2) ν .

Neumann series of Bessel functions

Recently, Pogány and Süli [Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 137(7) (2009), pp. 2363–2368] derived a closed-form integral expression for a Neumann series of Bessel functions. In this note, our aim is to establish another kind of integral representations for the Neumann series of Bessel functions of the first kind J ν.

Integral representation of first kind Kapteyn series

The main aim of this research note is to establish two different types of integral representation formulae for the Kapteyn series of the first kind. The first one is a double definite integral expression, while the second type includes indefinite integral representation formula.

ON COEFFICIENTS OF KAPTEYN-TYPE SERIES

Quite recently Jankov and Pogány [JANKOV, D.—POGÁNY, T. K.: Integral representation of Schlömilch series, J. Classical Anal. 1 (2012) 75–84] derived a double integral representation of the Kapteyn-type series of Bessel functions. Here we completely describe the class of functions Λ = {α}, which generate the mentioned integral representation in the sense that the restrictions

Integral representations of Dini series of Bessel functions

\alpha |_\mathbb{N} = (\alpha _n )_{n \in \mathbb{N}}

Two-Sided Inequalities for the Extended Hurwitz-Lerch Zeta Function

is the sequence of coefficients of the input Kapteyn-type series.

Integral representation of Schlömilch series

Two different type integral representation formulae are established for Dini (or Fourier–Dini) series of Bessel functions. The first one is a double definite integral expression derived by using a method successfully applied in the similar questions for Neumann, Kapteyn and Schlömilch series, while the another double indefinite integral formula is concluded by means of the non-homogenous Bessel differential equation.