Related papers: Relationship between Vieta-Lucas polynomials and L…

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n,…

We introduce a sequence $P_{2n}$ of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of $P_{2n}$ and its degree $d$ has a limit when…

We show that Wilson's theorem as well as the Wilson quotient can be described by supercongruences modulo any higher prime power involving terms of power sums of Fermat quotients. The new approach uses Bell polynomials and Newton's…

The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been…

The Gauss--Lucas theorem states that any convex set $K\subset\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\in\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following…

Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in…

Let $P_1,...,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,...,d_n$ respectively. We prove that if $\nu$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<\nu,$ then all multivariate subresultants…

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the…

In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial $p$ (i.e. a partially symmetric tensor of $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex,…

Lanczos methods for solving $\textit{A}\textbf{x}=\textbf{b}$ consist in constructing a sequence of vectors $(\textbf{x}_k), k=1,...$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}=\textit{P}_{k}(\textit{A})\textbf{r}_{0}$,,…

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $\left(U_n\right)_{n\geq0}$ and the second Lucas sequence $\left(V_n\right)_{n\geq0}$ are defined respectively by $U_{n+2}=aU_{n+1}+bU_{n},\, U_0=0,\,U_1=1$ and…

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are…

The paper presents analogues of some Vieta formulas for quaternionic polynomials of the form $R_n(w)=\sum\limits_{m=0}^{n}A_mx^m$. A criterion for the sphericity of the root (zero) of a polynomial $R_n(w)$ is established.

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

Motivated by problems of comparative genomics and paleogenomics, in [Chauve et al., 2009], the authors introduced the Gapped Consecutive-Ones Property Problem (k,delta)-C1P: given a binary matrix M and two integers k and delta, can the…

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic:…

New criteria are shown that certain combinations of finite unimodal polynomials are unimodal. %Given unimodal polynomials with explicit expressions and dependent recursion relations, we propose an approach to determine their modes. As…

The sequence $A(n)_{n \geq 0}$ of Ap\'ery numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear combination of…

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a…

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc,…