Related papers: Certain results on Euler-type integrals and their …

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of…

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex…

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

We show how the formulas in paper Variae considerationes circa series hypergeometricas written by Euler imply the duplication formula for the Gamma-function. This paper can be seen as an Addendum to a previous paper by the author.

This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers…

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The…

In this paper we discuss three types of the mean values of the Euler double zeta function. In order to get results we introduce three approximate formulas for this function.

Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.

We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…

We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the classical Euler integrals. The domains of integration of our so-called Euler--Mellin integrals are naturally related to the…

In this paper, we present an extension of Mittag-Leffler function by using the extension of beta functions (\"{O}zergin et al. in J. Comput. Appl. Math. 235 (2011), 4601-4610) and obtain some integral representation of this newly defined…

In this article we give a result obtained of an experimental way for the Euler totient function.

We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.

We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.

The electron repulsion integrals over the Slater-type orbitals with non-integer principal quantum numbers are considered. These integrals are useful in both non-relativistic and relativistic calculations of many-electron systems. They…

In this paper, we define a new type multivariable hypergeometric function. Then, we obtain some generating functions for these functions. Furthermore, we derive various families of multilinear and multilateral generating functions for these…

The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead…

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get…

In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function.…