Related papers: Mean values of long Dirichlet polynomials with div…
Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…
The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of…
We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the…
The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…
In this paper, the asymptotic formulas for Eulerian numbers, refined Eulerian numbers and the coefficients of descent polynomials are obtained directly from the spline interpretations of these numbers. Having related these numbers directly…
The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to…
We obtain new mean value theorems for exponential sums with very smooth numbers, which provide a power saving against the trivial bound in region where previous bounds do not apply.
We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arves\'u et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the…
We introduce and study the generalized cyclotomic polynomials $\Phi_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients,…
This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling…
The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This…
We give an analytic proof of the solution of Dirichlet Problem for continous functions satisfying a nonlinear mean value problem related to the p-laplace operator and certain stochastic games.
We prove an asymptotic formula for a variant of the binary additive divisor problem with linear factors in the arguments, which has a power saving error term and which is uniform in all involved parameters.
We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the…
Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being…
The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series…
We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as…
Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of…
We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…