Related papers: $L^\infty$-estimation of generalized Thue-Morse tr…

Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $\Lambda_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the…

We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence…

Let $q, r, s \geqslant 1$ satisfy $\frac{1}{2q} + \frac{1}{2r} = \frac{1}{s}$ and $X \in \mathcal{C}_s(\mathcal{H})$. If $(\lambda_n)_{n=1}^{\infty}, (w_n)_{n=1}^{\infty}$ are sequences in $(0,+\infty)$ and $(\lambda_n^{(1-q)/(2q)}…

Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( \xi_k \sin (kt) + \eta_k \cos (kt)\right), $$ where $(\xi_1,\eta_1),(\xi_2,\eta_2),\ldots$ are independent identically…

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*}…

For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this…

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$,…

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector…

We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right)…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

This contribution aims at studying the behaviour of the classical sample moment estimator, $S(n,q)= \sum_{k=1}^n X_k^{q}/n $, as a function of the number of available samples $n$, in the case where the random variables $X$ are positive,…

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $\nu$ can be uniquely…

The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here…

We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the…

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…

Let $\gamma_{d}$ be the $d$-dimensional standard Gaussian measure and $\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{1}(\gamma_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge 0}$ is equivalent to the…

Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading…

It is shown in [7] by Venkaiah in 2015 that a category of the number of generalized can be computed using the expression \begin{equation*} e(n, q) = \frac{1}{(q-1) ord(\lambda) n} \sum^{ord(\lambda)n}_{\substack{t \in \mathbb{F}_q \setminus…

We examine the four F\'ejer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<\pi)\] where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The analysis of the sums with $f(x)=g(x)=\cos x$,…

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and…