Related papers: New definite integrals and a two-term dilogarithm …

The integral $\int_{|z|=1} \frac{z^\beta}{z-\alpha} dz$ for $\beta=\frac{1}{2}$ has been comprehensively studied by Mortini and Rupp for pedagogical purposes. We write for a similar purpose, elaborating on their work with the more general…

Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*}…

By means of an $L^1-L^\infty$ duality argument, it is proved that, in a suitable planar domain $\Omega$, the solution $u$ to the IBVP associated to the Navier-Stokes equations, with initial datum $u_0\in L^2(\Omega)$, satisfies the…

The family of Cauchy transforms \[C_{g}(z,w) = -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-z) } da(u ),\] where the measurable function $g$ with compact (essential) support satisfies $0 \leq g\leq 1,$ and suitably defined…

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity…

We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are…

We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional $I(u) = \int |\nabla u|^2 + V(u)$, where $V(u)$ is the characteristic function of the interval $(-1,1)$. This functional is a…

The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was…

We prove two identities of Hall-Littlewood polynomials, which appeared recently in a paper by two of the authors. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition…

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a…

We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3\pi}2.$$ Motivated by Galois…

We obtain an asymptotic expansion for the sum \[S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue…

For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $\alpha\beta$. We obtain a theoretical representation of such $K_1$…

In this paper, we evaluate some series via the WZ method, and confirm several previous conjectures. For example, we prove the following two identities conjectured by the second author: $$\sum_{k=0}^{\infty} \frac{(28k^2 + 10k + 1)…

We consider two-dimensional Coulomb systems confined in a disk with ideal dielectric boundaries. In particular we study the two-component plasma in detail. When the coulombic coupling constant $\Gamma=2$ the model is exactly solvable. We…

We investigate here the nonlinear elliptic H\'enon type equation: $$\D^{2} u= |x|^a|u|^{p-1}u \; \,\,\mbox{in}\,\,\,\, \R^{n}_{+}, \quad \quad u =\frac{\partial u}{\partial x_n} = 0 \quad \mbox{in}\,\,\,\, \partial \R^{n}_{+},$$ with $p>1$…

We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q} \frac{\dd t}{\pi( t^2+1)}.$$ These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and…

In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The…

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to…

Let $\K$ be the complete elliptic integral of the first kind. In this paper, the authors prove that the function $r\mapsto r^{-2}\{[\log(2\K(r)/\pi)]/\log((\arth r)/r)-3/4\}$ is strictly increasing from $(0,1)$ onto $(1/320,1/4)$, so that…