Related papers: On Bergeron's positivity problem for $q$-binomial …

Let k and n be positive integers. We mainly show that $$(ln+1) | k\binom{kn+ln}{kn},$$ $$2\binom{kn}n | \binom {2n}{n}C_{2n}^{(k-1)}$$, $$\binom{kn}n | (2k-1)C_n\binom{2kn}{2n},$$ $$\binom{2n}n | (k+1)C_n^{(k-1)}\binom{2kn}{kn},$$…

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic…

We prove strict unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of S_n representations.

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for…

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace…

In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated…

We consider the set of monic real univariate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all…

We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the…

After a brief review of existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then…

The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing $B_{r}$ by…

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…

The $q$-binomial coefficients are q-analogues of the binomial coefficients, counting the number of $k$-dimensional subspaces in the $n$-dimensional vector space $\mathbb{F}^n_q$ over $\mathbb{F}_{q}$. In this paper, we define a Euclidean…

The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(\alpha)$…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of…

In this paper, we present several new $q$-congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. As a conclusion, we obtain the following congruence: \begin{align*}…

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

We develop a new kind of nonnegativity certificate for univariate polynomials on an interval. In many applications, nonnegative Bernstein coefficients are often used as a simple way of certifying polynomial nonnegativity. Our proposed…