Related papers: New series representation for Madelung constant
A direct summation method for the Madelung constant calculation is presented where a crystal lattice is constructed from linear arrays of charges or axial multipoles. An array is designed to have vanishing low order electric moments such…
A new method of computing the Madelung constants for hypercubic crystal structures in any dimension $n\geq 2$ is given. It is shown for $n\geq 3$ that the Madelung constant may be obtained in a simple, efficient and unambiguous way as the…
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the…
We suggest a new approach to Artin's constant that leads to its representation as an infinite sum divided by another infinite sum. The same approach works well for Stephens' constant and higher rank Artin's constants. The main results are…
We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung…
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
In this paper we derive rapidly converging series for Catalan's constant and for Ap\'ery's constant. The method may be easily generalised to produce new series representations for other values of the Riemann zeta function and the Dirichlet…
An efficient real space method is derived for the evaluation of the Madelung's potential of ionic crystals. The proposed method is an extension of the Evjen's method. It takes advantage of a general analysis for the potential convergence in…
By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we…
An efficient real space method is derived for the evaluation of the Madelung's potential of ionic crystals. The proposed method is an extension of the Evjen's method. It takes advantage of a general analysis for the potential convergence in…
We present a new representation of the Stieltjes constants in the form of a limit of a Fourier series.
Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly…
In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
The Stieltjes constants \gamma_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \zeta(s,a) about s=1. We present series representations of these constants of interest to theoretical…
We propose a simple direct-sum method for the efficient evaluation of lattice sums in periodic solids. It consists of two main principles: i) the creation of a supercell that has the topology of a Clifford torus, which is a flat, finite and…
We provide representations of Euler's constant $\gamma=0.577...$ as series which converge geometrically fast (but use coefficients whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are discussed.
In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical…
The Coulomb potential at an interior ion in a finite crystal of size $p$ is given by a linear superposition of contributions from displacement vectors ${\mathbf r}=(x,y,z)$ to its neighbors. This additive structure underlies universal…