Related papers: A q-rious positivity
q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric…
We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish…
The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$…
This is an informal and mostly expository note describing some asymptotic behavior and qualitative properties of the q-binomial coefficients. The results are mostly not new, but the overall story we present does not seem to be well known --…
In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula…
We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…
Bergeron--Ceballos--K\"ustner introduced the $q$-Fibonomial coefficients \( \qfibonom{m+n}{n}\), gave a combinatorial interpretation of the $q$-Fibonomial coefficients via a weighted path-domino tiling model, and conjectured that these…
For the quantum integer $[n]_q = 1+q+...+q^{n-1}$ there is a natural polynomial multiplication $*_q$ such that $[m]_q *_q [n]_q = [mn]_q$. This multiplication leads to the functional equation $f_{mn}(q) = f_m(q)f_n(q^m),$ defined on a given…
Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…
We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\genfrac{[}{]}{0pt}{}{n}{k}_{q}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more…
In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and…
We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over…
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…
We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the…
We prove that $$ \prod_{n=0}^{\infty}(1+q^{10n+5}) = \frac{\sum_{n=-\infty}^{\infty}q^{n^{2}}\, \sum_{n=-\infty}^{\infty}(-1)^{n}\, (-q)^{n(3n-1)/2}}{4\, \sum_{n\geq 0}(-q)^{\frac{n(n+1)}{2}}\sin \left\{\frac{(2n+1)3\pi}{10}\right\}\,…
We show that Thom polynomials of Lagrangian singularities have nonnegative coefficients in the basis consisting of Q-functions. The main tool in the proof is nonnegativity of cone classes for globally generated bundles.
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…
Let m_1,...,m_s be positive integers. Consider the sequence defined by multinomial coefficients: a_n=\binom{(m_1+m_2+... +m_s)n}{m_1 n, m_2 n,..., m_s n}. Fix a positive integer k\ge 2. We show that there exists a positive integer C(k) such…
Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…
$q$-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, $q$-analogs of various probability distributions have been…