Related papers: A q-rious positivity

The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a…

Given a nonnegative integer $m$ and a finite collection ${\mathcal A}$ of linear forms on ${\mathbb Q}^d$, the arrangement of affine hyperplanes in ${\mathbb Q}^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in {\mathcal A}$ and…

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…

Let $q$ be a positive integer. Recently, Niu and Liu proved that if $n\ge \max\{q,1198-q\}$, then the product $(1^3+q^3)(2^3+q^3)\cdots (n^3+q^3)$ is not a powerful number. In this note, we prove that (i) for any odd prime power $\ell$ and…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

In this paper, we investigate some properties of q-Bernoulli polynomi- als arising from q-umbral calculus. Finally, we derive some interesting identities of q-Bernoulli polynomials from our investigation.

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of…

We obtain some properties of the q-Narayana polynomials for q=-1 and compare them to corresponding properties for q=1.

Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers…

The celebrated (First) Borwein Conjecture predicts that for all positive integers~$n$ the sign pattern of the coefficients of the ``Borwein polynomial'' $$(1-q)(1-q^2)(1-q^4)(1-q^5) \cdots(1-q^{3n-2})(1-q^{3n-1})$$ is $+--+--\cdots$. It was…

Let $q=4$ and $k$ a positive integer. In this short note, we present a class of permutation polynomials over $\Bbb F_{q^{3k}}$. We also present a generalization.

We say a polynomial f having integer coefficients is strongly coefficient convex if the set of coefficients of f consists of consecutive integers only. We establish various results suggesting that the divisors of x^n-1 with integer…

The aim of this paper is to give an example of a non-commutative discrete hypergroup associated with $q$-disk polynomials. These are polynomials $R_{l,m}^{(\a)}$ in two non-commuting variables which are expressed through little $q$-Jacobi…

For a real number $q>1$ and a positive integer $m$, let $Y_m(q):={\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ In this paper, we show that $Y_m(q)$ is dense in ${\Bbb R}$ if and only if $q<m+1$ and $q$…

The q-binomial coefficients were assumed to be unimodal as early as the 1850's, but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an "elementary" proof using linear algebra. Finally, in…

In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.