Related papers: Convergence acceleration of series through a varia…

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…

We propose a new simple convergence acceleration method for wide range class of convergent alternating series. It has some common features with Smith's and Ford's modification of Levin's and Weniger's sequence transformations, but its…

The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…

We show how to convert divergent series, which typically occur in many applications in physics, into rapidly convergent inverse factorial series. This can be interpreted physically as a novel resummation of perturbative series. Being…

Martingale methods are used to study the almost everywhere convergence of general function series. Applications are given to ergodic series, which improves recent results of Fan \cite{FanETDS}, and to dilated series, including Davenport…

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…

We investigate a class of combinatorial sums involving reciprocals of central binomial coefficients , employing generating functions as the primary solution technique to formulate and analyze series involving the Catalan's constant. Using a…

In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…

Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…

In our recent publications we have introduced the incomplete cosine expansion of the sinc function for efficient application in sampling [Abrarov & Quine, Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7 (2)…

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the…

In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series,…

To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann…

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

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…

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same…

In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…

A new representation for a regular solution of the radial Dirac system of a special form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to the spectral parameter. For the…