Related papers: On Finite Pseudorandom Binary Sequences: Generaliz…

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…

In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…

In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in…

An M-sequence generated by a primitive polynomial has many interesting and desirable properties. A pseudo-random array is the two-dimensional generalization of an M-sequence. There are non-primitive polynomials all of whose non-zero…

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with…

Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials.…

This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are…

We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a…

We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type.…

In this paper we present the notions of trail (pseudo-)division, generalized subresultants and generalized subresultant algorithm.

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially…

We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating…

We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We…

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this…

Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically.

In this paper we develop a very general class of bivariate discrete distributions. The basic idea is very simple. The marginals are obtained by taking the random geometric sum of a baseline distribution function. The proposed class of…

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…