Related papers: Generalized Pascal triangle for binomial coefficie…
In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…
We introduce the $k$-bonacci polyominoes, a new family of polyominoes associated with the binary words avoiding $k$ consecutive $1$'s, also called generalized $k$-bonacci words. The polyominoes are very entrancing objects, considered in…
We construct a word-theoretic framework for generalized Markov numbers, that is, positive integers appearing in positive integer solutions of the generalized Markov equation $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$. For each…
Pascal's triangle is widely used as a pedagogical tool to explain the "first-order" multiplet patterns that arise in the spectra of $I_N S$ coupled spin-1/2 systems in magnetic resonance. Various other combinatorial structures, which may be…
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) non trivial central extension of the Heisenberg algebra. Using the boson representation of the latter, we construct the corresponding polynomial analogue…
In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…
We give a generalization of Collatz conjecture or 3n+1 problem on 2-adic completion of Q. A isometric of $Q_2$ provides information on the average behavior of the firsts terms of the sequence according to the class of $u_0$ modulo $2^m$. A…
This paper studies the product $\bar{G}_n$ of the binomial coefficients in the n-th row of Pascal's triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order n. It studies its size as a…
We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials…
We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…
For $p$ prime, let $\mathcal{H}^n$ be the linear span of characteristic functions of hyperplanes in $(\mathbb{Z}/p^k\mathbb{Z})^n$. We establish new upper bounds on the dimension of $\mathcal{H}^n$ over $\mathbb{Z}/p\mathbb{Z}$, or…
The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $\Gamma^{(p)}_n$, which are subgraphs of $Q_n$ induced by…
It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
The locality of words is a relatively young structural complexity measure, introduced by Day et al. in 2017 in order to define classes of patterns with variables which can be matched in polynomial time. The main tool used to compute the…
A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and…
As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted…
An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one…
The generalized complex numbers can be realized in terms of $2\times2$ or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of…