Related papers: Laurent Phenomenon Sequences

In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the…

We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…

One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the "Laurent phenomenon").…

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan…

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula…

We show that Laurent biorthogonal polynomials whose defining three-term recurrence have constant coefficients have coefficient arrays that are Riordan arrays. For each such family of Laurent biorthogonal polynomials we associate in a…

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue…

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky.…

We classify all linear division sequences in the integers, a problem going back to at least the 1930s. As a corollary we also classify those linear recurrence sequences in the integers for which $(x_m,x_n)=\pm x_{(m,n)}$. We also show that…

Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each…

Laurent phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent phenomenon algebras, defined by Lam and Pylyavskyy,…

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-\beta. We combine both lattices to obtain the bi-lattice N \cup (N+1-\beta) and show that the orthogonal polynomials…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this…

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by…

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…

Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…