Related papers: Congruences for generalized Fishburn numbers at ro…

Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of…

We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a…

For every genus $g\geq 2$, we construct an infinite family of strongly quasipositive fibred knots having the same Seifert form as the torus knot $T(2,2g+1)$. In particular, their signatures and four-genera are maximal and their homological…

We estimate the covariance in counts of almost-primes in $\mathbb{F}_q[T]$, weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This generalizes recent work of Keating and Rudnick that estimates the…

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…

Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb R}^{+}$ a log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let $$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank}(x)} \in {\mathbb…

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$…

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a…

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound…

We study special values of finite multiple harmonic q-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the…

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence…

A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…

In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson…

Let $p$ and $q$ be distinct prime numbers, with $q\equiv 1\pmod{12}$. Let $N$ be a positive integer that is coprime to $pq$. We prove a formula relating the Hasse--Weil zeta function of the modular curve $X_0(qN)_{\mathbb{F}_q}$ to the…

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian…

We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a…

We continue the first and second authors' study of $q$-commutative power series rings $R=k_q[[x_1,\ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{\pm 1}_1,\ldots,x^{\pm 1}_n]]$, specializing to the case in which the commutation parameters…

A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions…

We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt--Clausen theorem and Kummer's…