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Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a…

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev-Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they…

Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal L $\infty$-and L 1-norm on K, respectively. This articles presents numerical procedures based on semidefinite programming to compute these…

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform…

In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…

We introduce a new problem on the elementary symmetric polynomials $\sigma_k$, stemming from the constraint equations of some modified gravity theory. For which coefficients is a linear combination of $\sigma_k$ $1/p$-concave, with $0 \leq…

C.M. Bender and G. V. Dunne showed that linear combinations of words $q^{k}p^{n}q^{n-k}$, where $p$ and $q$ are subject to the relation $qp - pq = \imath$, may be expressed as a polynomial in the symbol $z = \tfrac{1}{2}(qp+pq)$. Relations…

Markov polynomials are the Laurent-polynomial solutions of the generalised Markov equation $$X^2 + Y^2 + Z^2 = kXYZ, \quad k=\frac{x^2 + y^2 + z^2}{x y z}$$ which are the results of cluster mutations applied to the initial triple $(x, y,…

Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$, $\sum_{6\mid k}\binom…

We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that…

With this article, we hope to launch the investigation of what we call the real zero amalgamation problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an…

For each positive integer n, we define a polynomial in the variables z_1,...,z_n with coefficients in the ring $\mathbb{Q}[q,t,r]$ of polynomial functions of three parameters q, t, r. These polynomials naturally arise in the context of…

Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…

Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and Williams. In this paper, properties of the binomial-Eulerian polynomials, including recurrence relations and generating functions are studied. We present three…

In this paper we study the distribution of consecutive square-free numbers of the forms $x^2+y^2+z+1$, $x^2+y^2+z+2$ and $x^2+y^2+z^2+z+1$, $x^2+y^2+z^2+z+2$, respectively. We establish asymptotic formulas for each of these two cases.

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…

We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and…