Related papers: Some definite logarithmic integrals from Euler sum…
By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta…
In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these…
For real numbers $p,q>1$ we consider the following family of integrals: \begin{equation*} \int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad \mbox{and}\quad…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.
Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of…
We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect…
Functional methods can be applied to the quantum effective action to efficiently determine counterterms and matching conditions for effective field theories. We extend the toolbox to two-loop order and beyond and show how to evaluate the…
Taking inspiration from the work of Lanphier \cite{LANPHIER2022125716}, a generalized multivariable polynomial formulation for sums of alternating powers is given, as well as analogous sums. Furthermore, an analog of the Euler-Maclaurin…
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact…
Integrals occurring in Thomas-Fermi theory which contains the logarithm of the Airy function Ai'(x) have been obtained in terms of analytical expressions.
We show the recurrence relations of the Euler-Zagier multiple zeta-function which describes the $r$-fold function with one variable specialized to a non-positive integer as a rational linear combination of $(r-1)$-fold functions, which…
In this paper we present several new classes of logarithmically completely monotonic functions. Our functions have in common that they are defined in terms of the $q-$gamma and $q-$digamma functions. As an applications of this results, some…
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.
A certain dilogarithmic integral I_7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an…
We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Gamma(q) and log sin(q).