Tibor K. Pogány

Integral and computational representations of the extended Hurwitz–Lerch zeta function

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

Integral representation of Mathieu (a, λ)-series

Integral representation and bilateral bounding inequalities are obtained for the Mathieu (a, λ)-series S(ϱ, μ, a, λ).

INTEGRAL REPRESENTATION FOR NEUMANN SERIES OF BESSEL FUNCTIONS

A closed integral expression is derived for Neumann series of Bessel functions ― a series of Bessel functions of increasing order ― over the set of real numbers.

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.

On multiple generalized Mathieu series

Closed integral form expressions and different kinds of bounding inequalities are obtained for the m-fold generalized Mathieu series where s, q are positive m-tuples that ensure the convergence of Mathieu series. Connections to the first kind Bessel functions are given.

Laplace type integral expressions for a certain three-parameter family of generalized Mittag–Leffler functions with applications involving complete monotonicity

Abstract In this paper, we derive a Laplace type integral expression for the function e α , β γ ( t ; λ ) defined by e α , β γ ( t ; λ ) ≔ t β − 1 E α , β γ ( − λ t α ) , where E α , β γ ( z ) stands for the generalized three-parameter Mittag–Leffler function occurring in many interesting applied problems involving fractional differential equations. Our result is shown to enable us to extend certain findings by Mainardi (2010) [21] and others. As an application of the obtained Laplace type integral representation, we prove the complete monotonicity of the function e α , β γ ( t ; λ ) . We also establish several related positivity results and some uniform upper bounds for the function e α , β γ ( t ; λ ) .

New integral forms of generalized Mathieu series and related applications

The main object of this article is to present a systematic study of integral representations for generalized Mathieu series and its alternating variant, and to derive a new integral expression for these special functions by contour integration using rectangular integration path. Also, by virtue of newly established integral form of generalized Mathieu series, we obtain a new integral expression for the Bessel function of the first kind of half integer order, solving a related Fredholm integral equation of the first kind with nondegenerate kernel.

A linear ODE for the Omega function associated with the Euler function Eα(z) and the Bernoulli function Bα(z)

Abstract The authors derive a linear ODE (ordinary differential equation) whose particular solution is the Butzer–Flocke–Hauss complete real-parameter Omega function Ω ( w ) , which is associated with the complex-index Bernoulli function B α ( z ) and with the complex-index Euler function E α ( z ) . This is accomplished here with the aid of an integral representation of the alternating Mathieu series S ˜ ( w ) . A new integral representation and some two-sided bounding inequalities are also given for the Omega 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 .

On an identity for zeros of Bessel functions

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.