Related papers: On a new general method of summation
In this paper we present a new mathematical conception based on a new method for ordering the integers. The method relies on the assumption that negative numbers are beyond infinity, which goes back to Wallis and Euler. We also present a…
We present a common ground for infinite sums, unordered sums, Riemann/Lebesgue integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
We propose an ensemble algorithm, which provides a new approach for evaluating and summing up a set of function samples. The proposed algorithm is not a quantum algorithm, insofar it does not involve quantum entanglement. The query…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
A new understanding of the notion of regularizer is proposed. It is argued that this new notion is more realistic than the old one and better fits the practical computational needs. An example of the regularizer in the new sense is given. A…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities,…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…