In our previous report [5], we developed some methods ie the study of the line-sequential properties of the polynomial sequences treated in Shannon and Horadam [9]. In this report, we work out the properties for the general case and apply them to the Pell polynomial line-sequence as an example. Some known results are included for completeness, but only new aspects will be presented in some deta...

The present paper deals with functions acting on the digits of an q-ary expansion. In particular let n be a positive integer, then we call

The multivariate orthogonal polynomials are related to a family of commuting selfadjoint operators. The spectral theorem for these operators is used to prove that a polynomial sequence satisfying a vector-matrix form of the three-term relation is orthonormal with a determinate measure.

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of ir...

Let q ≥ 2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the repartition of the sequence (sq(P (n)))n∈N in arithmetic progressions when P ∈ Z[X] is such that P (N) ⊂ N. We answer Gelfond’s question and we show the uniform distribution modulo 1 of the sequence (αsq(P (n)))n∈N for α ∈ R \Q...

It is well known that the coefficients of matching polynomial are unimodal. Unimodality (or their absolute values) other graph polynomials has been studied as well. One way to prove unimodality real-rootedness. Recently I. Beaton and J. Brown (2020) proved for almost all graphs domination form a unimodal sequence, C. Barton, D. Pike forest (aka acyclic polynomial) real-rooted iff G forest. Let ...

We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of...

In this paper we develop methods to provide unified approaches to the reality of zeros of polynomial sequences satisfying certain recurrence relations and then apply them to derive some known results and solve certain open problems and conjectures.

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in L to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilman...