Related papers: Strong arithmetic property of certain Stern polyno…

For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new…

For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that…

Let $s$ denote West's stack-sorting map. For all positive integers $m$ and all integers $n\geq 2m-2$, we give a simple characterization of the set $s^{n-m}(S_n)$; as a consequence, we find that $|s^{n-m}(S_n)|$ is the $m^\text{th}$ Bell…

In this paper, we find an elementary approach for double sums where the inner sum is binomial but incomplete. We apply our core identity and its relatives to double sums involving famous numbers such as harmonic numbers, Fibonacci numbers,…

Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$ variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element subsets of…

We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a…

For each algebraic number $\alpha$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(\alpha)$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at…

Given a polynomial $Q(x_1,\cdots, x_t)=\lambda_1 x_1^{k_1}+\cdots +\lambda_t x_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv…

We show that if m is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial P_n(z;B) solves an explicit second order linear…

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$)…

We show that the Schur polynomials in all primitive $n$th roots of unity are $1$, $0$, or $-1$, if $n$ has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the…

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $\phi$, all in…

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots,…

For $n \times n$ matrices $A$ and $B$ define $$\eta(A,B)=\sum_{S}\det(A[S])\det(B[S']),$$ where the summation is over all subsets of $\{1,..., n\}$, $S'$ is the complement of $S$, and $A[S]$ is the principal submatrix of $A$ with rows and…

For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the…

The multi-variable Schmidt polynomials are defined by $$ S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k. $$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients…

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits…