Related papers: On Lehmer's question for integer-valued polynomial…
In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between…
We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In…
In this survey article, we review some results and conjectures related to orthogonal polynomials on Cantor sets. The main purpose of this paper is to emphasize the role of equilibrium measures in order to have a general theory of…
In this paper we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev…
We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution…
In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate…
The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the…
Inspired by the work of Deninger, we present a formula that relates the Mahler measure of a two-variable variant of cyclotomic polynomial to regulator of class in motivic cohomology associated to cyclotomic fields and linear combination of…
In this paper we investigate some properties for the q-Euler numbers ans polymials. From these properties we give some identities on the Bernstein polymials and q-Euler polynpmials.
We consider inequalities of Bombieri type for polynomials that need not be homogeneous, using the apolar inner product.
In the paper, we generalize some congruences of Lehmer for general composite numbers.
We describe some studies related to the frequency of prime values of integer polynomials.
Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We consider upper and lower bounds on the minimal height of an irrational number lying in a particular real quadratic field.
We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…
In the present paper a new mean value theorem for polynomials of special form is obtained. The case of sums on vertices of a regular polygon is studied. A criterion for a certain equation to be satisfied is obtained.
We derive Mehler--Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $x=0,$ and determine the…
In this paper, we prove an inequality regarding the differential polynomial. This improves some recent results.
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.