Related papers: Special functions, integral equations and Riemann-…

Let $G=(V, E)$ be a connected finite graph, $h$ be a positive function on $V$ and $\lambda _{1}(V)$ be the first non-zero eigenvalue of $-\Delta$. For any given finite measure $\mu$ on $V$, define functionals \begin{eqnarray*} J_{ \beta…

Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on an…

Based on $k$-gamma and $k$-digamma functions, we show four series expansions to the Furdui-type integral related to Riemann zeta function and hypergeometric function, and also present some new identities, series expansions and inequalities…

Let $\alpha,\beta$ be real parameters and let $a>0$. We study radially symmetric solutions of \begin{equation*} S_k(D^2v)+\alpha v+\beta \xi\cdot\nabla v=0,\, v>0\;\; \mbox{in}\;\; \mathbb{R}^n,\; v(0)=a, \end{equation*} where $S_k(D^2v)$…

The vector Riemann-Hilbert problem is analyzed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros…

The purpose of this paper is to study the solutions of $$ \Delta u +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$\alpha_p(K)=\sup\left\{\alpha \in \mathbb{R}:\, \int_{\mathbb{R}^2}…

One-dimensional and two-dimensional integrals containing $E_b(-u)$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ are considered. $E_b(-u)$ is the Mittag-Leffler function and the integral is taken over the rectangle $0 \leq x <…

We consider the singular perturbation problem $$ \Delta u_\epsilon=\beta_\epsilon(u_\epsilon), $$ where $\beta_\epsilon(s)=\frac{1}{\epsilon}\beta(\frac{s}{\epsilon})$, $\beta$ is a Lipschitz continuous function such that $\beta>0$ in $(0,…

The initial value problem and global properties of solutions are studied for the vector equation: $\Big(\|u'\|^{l}u'\Big)'+\|A^{\frac{1}{2}}u\|^\beta Au+g(u')=0$ in a finite dimensional Hilbert space under suitable assumptions on $g$.

We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A…

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…

The purpose of this article is two-fold. First, we investigate the inequality $$ -\Delta u+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we…

In this paper, we study the uniform H\"older continuity of the generalized Riemann function $R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad…

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses…

Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are finite indices $\delta_i, \beta_i, i=1,2$ and a function $u$, all of which are defined in terms of the characteristics of $X_t$, such that \liminf_{t\to0}u(t)^{-1/\eta}X_t^*=…

By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their…