Related papers: On Calculating the Coefficients of a Polynomial Ge…

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…

We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on…

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…

Given finitely many consecutive terms of an infinite sequence, we discuss the construction of a polynomial difference equation that the sequence may satisfy. We also present a method to seek a candidate polynomial differential equation for…

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties…

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the…

In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference…

We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…

Using a generating function approach, the correlation coefficients of four different graph-theoretical indices, namely the number of independent vertex subsets, the number of matchings, the number of subtrees and the Wiener index, are…

We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are…

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders.…

We study certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigate their number of real solutions. Our main focus is set on pairs of plane algebraic curves which form a so-called…

The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\inZ and n\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish…

We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…

A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…

Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize…

In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we…

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a…

We count the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This is equivalent to counting codewords in Reed-Solomon codes which are at a certain distance from a received…

The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the…