Related papers: Infinite divisibility of Smith matrices

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a…

It is shown that for positive real numbers $ 0<\lambda_{1}<\dots<\lambda_{n}$, $\left[\frac{1}{\beta({\lambda_i}, {\lambda_j})}\right]$, where $ \beta(\cdot,\cdot)$ denotes the beta function, is infinitely divisible and totally positive.…

Let $PD(\mathbb{R})$ be the family of continuous positive definite functions on $\mathbb{R}$. For an integer $n>1$, a $f\in PD(\mathbb{R})$ is called $n$-divisible if there is $g\in PD(\mathbb{R})$ such that $g^n=f$. Some properties of…

Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integer…

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ with at most one 1 in each row and each column such that…

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any prime that…

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\…

Let $p_1<p_2<\cdots<p_n$ be positive real numbers. It is shown that the matrix whose $i,j$ entry is $(p_i+p_j)^{p_i+p_j}$ is infinitely divisible, nonsingular and totally positive.

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…

Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and…

The matrix $S = [1+x_i y_j]_{i,j=1}^{n}, 0<x_1<\cdots<x_n,\, 0<y_1<\cdots<y_n$, has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of $S$ in terms of elementary…

For given integers a,b, and j at least 1 we determine the set of integers n for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite; we find explicitly the exceptional cases for which a,b the set is finite. For j=2,…

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues $\lambda_j$, we set…

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest…

It is well known that a non-constant complex-valued function $f$ defined on the open unit disk $\mathbb D$ is an analytic self-mapping of $\D$ if and only if Pick matrices $\left[…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup…