Related papers: Integrality and the Laurent phenomenon for Somos 4…
LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a…
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness…
We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the…
We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation…
In this paper, a simple proof that shows that the Somos-5 sequence produces only natural numbers is given. We also give simple proof that each $a_n$ in the Somos-5 sequence is relatively prime to its four predecessors. The manner in which…
We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This…
The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to…
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…
For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$,…
We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(\Sigma)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the…
A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…
The $LS$-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone, who also proved that under an appropriate choice of the parameters $L$ and $S$, such sequences are low-discrepancy. The…
We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general…
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…
The Burchnall-Chaundy polynomials $P_n(z)$ are determined by the differential recurrence relation $$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$ with $P_{-1}=P_0(z)=1.$ The fact that this recurrence relation has all solutions…