We study moments of the difference $D_n\left(x\right)-x^{n}n!{\mathrm{e}}^{-1/x}$ concerning derangement polynomials $D_n\left(x\right)$. For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x>0$. For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Let $\left[t\right]$ be the integral part of a real number $t$, and let $f$ be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum $S_f\left(x\right)={\sum }_{n\le x}f\left([x/n]\right)$, which improves the recent result of J. Stucky (2022).

Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $C_p\left(X\right)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p\left(X\right)$. We show that every exponentially $\kappa$-cofinal space $X$ has a ${\kappa }^{+}$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw\left(X\right)\le \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal...

Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,...,x_m$. Suppose $G$ is a subgroup of $S_m$, and $\chi$ is an irreducible character of $G$. Let $H_d(G,\chi )$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $\chi$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{\chi }\left(T\right)\in \mathrm{End}\left(H_d(G,\chi )\right)$ acting on symmetrized decomposable polynomials by $$K_{\chi }\left(T\right)(f_1*f_2*...*f_d)=T{f}_1*T{f}_2*...*T{f}_d.$$ In this paper, we show that the representation $T\mapsto K_{\chi }\left(T\right)$ of the general linear group $GL\left(V\right)$ is equivalent to the direct sum of $\chi \left(1\right)$ copies...

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results...

Let $𝔻$ denote the unit disk $\{z:|z|<1\}$ in the complex plane $ℂ$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline{𝔻}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^{\text{'}}$  has exactly one critical point outside $\overline{𝔻}$.

Let $k\ge 2$ and let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence defined by $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}$$ for $n\ge 2$ with initial conditions $$P_{-(k-2)}^{\left(k\right)}=P_{-(k-3)}^{\left(k\right)}=\cdots =P_{-1}^{\left(k\right)}=P_0^{\left(k\right)}=0,P_1^{\left(k\right)}=1.$$ In this study, we handle the equation $P_n^{\left(k\right)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{\left(k\right)}=y^m$ with $2\le y\le 1000$ has only one solution given by $P_7^{\left(2\right)}={13}^2.$

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation ${(-1)}^{u}r{a}^x+{(-1)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a\&gt;1$, $b\&gt;1$, $c\&gt;0$, $r\&gt;0$ and $s\&gt;0$. When $gcd(ra,sb)=1$, we show that $N\le 3$ except for a finite number of cases all of which satisfy $max(a,b,r,s,x,y)\&lt;2·{10}^{15}$ for each solution; when $gcd(a,b)\&gt;1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.

We show that the discriminant of the generalized Laguerre polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n\ge 1$, if and only if $(n,\alpha )$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is the alternating group $A_n$. For example, we establish that for all but finitely many positive integers $n\equiv 2(mod4)$, the only $\alpha$ for which the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is $A_n$ is...

We prove that almost all positive even integers $n$ can be represented as $p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{4}N|\le N^{1-1/54+\epsilon }$ for $2\le k\le 5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_1+p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{5}N|\le N^{1-1/54+\epsilon }$ for $1\le k\le 5$.

We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$ $\epsilon ,$ $k$ and $\beta .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $$\sum _{1\le h\le H}\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{n\in {𝒬}_k}}e\left(\vartheta hn\right).$$

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for...