Related papers: A class of logarithmic integrals

We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1) and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.

We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand -- a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In…

The aim of this paper is to prove the following result. For any commutative formal group ${\frak F}(x\otimes 1,1\otimes x),$ which is considered as a formal group over $H_\mathbb{Q},$ there exists a homomorphism to a formal group of the…

We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining…

We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general $q^2$ are…

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period…

This is an announcement of certain rationality results for the critical values of the degree-2n L-functions attached to GL(1) $\times$ SO(n, n) over $\mathbb Q$ for an even positive integer n. The proof follows from studying the rank-one…

Our goal in this work is to found a closed form for rational generat- ing functions, these generate a various families of polynomials and generalized polynomials, in order to get the general recursive formula satisfied by these polynomials.

The rational Landen transformation is a map on the space of coefficients of a rational integrand that preserves the value of the integral. We provide a family of these transformations that apply to rational integrands on the whole line.…

We prove, in a quantitative form, linear independence results for values of a certain class of q-series, which generalize classical q-hypergeometric series. These results refine our recent estimates.

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic…

We introduce the continued logarithm representation of real numbers and prove results on the occurrence and frequency of digits with respect to this representation

We make some observation on the logarithmic version of K-stability.

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…

We classify the completely integrable systems associated with classical root systems whose potential functions are meromorphic at an infinite point.

The logarithmic integral no. 4.325.7 from Gradshteyn and Ryzhik's tables of integrals was first evaluated by Malmst\'en. Recently, Blagouchine used contour integration methods to evaluate a family of logarithmic integrals that contains this…

In this paper, we answer the question: "what is the qth Fibonacci number, where q is a positive rational?". The answer is the codenominator function, which is an integral-valued map. It is defined via a pair of functional equations. Many…

We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in…

An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive…