New Notation in Series of Functions

Miskolc Mathematical Notes, 2019

Communications of the Korean Mathematical Society, 2003

Formal manipulations of double series are useful in getting some other identities from given ones and evaluating certain summations, involving double series. The main object of this note is to summarize rather useful double series manipulations scattered in the literature and give their generalized formulas, for convenience and easier reference in their future use. An application of such manipulations to an evaluation for Euler sums (in itself, interesting), among other things, will also be presented to show usefulness of such manipulative techniques.

Http Digital Bl Fcen Uba Ar, 2011

has received special interest from the applications. In some way, this representations resemble the classic Karhunen-Loève theorem [27]. A property of the Karhunen-Loève expansion of a random process is that one obtains an orthonormal basis of the closed linear span of the whole process. This allows to write certain approximations as unconditional convergent series. This useful property could be obtained under other conditions. To solve this problem, nally, we study conditions under which a stationary sequence forms a frame or a Riesz basis of its closed linear span.

arXiv (Cornell University), 2012

A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations of the notation include definitions of constant points, periodic points, a filled-in Julia set, the Mandelbrot set, iterations of a logistic map, the double-approximating procedure for solving the Lorenz equations, a description of a financial time series, and reordering nonnegative integers useful for the investigation of the Collatz's (3x+1)/2 convergence problem. The terms iteral and iteral of function are suggested to name the new denomination.

2003

The use of registered names. trademarks etc. in this publication does not imply. even in the absence of a specific statement. that such names are exempt from the relevant laws and regulations and therefore free for general use.

Springer eBooks, 2016

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

A remarkably large number of operational techniques have drawn the attention of several researchers in the study of sequences of functions and polynomials. In this sequel, here, we aim to introduce a new sequence of functions involving a product of the generalized Mittag-Leffler function by using operational techniques. Some generating relations and finite summation formula of the sequence presented here are also considered.

Applied Mathematics Letters, 2008

The present work presents some necessary and sufficient conditions for the convergence to a periodic function of a special kind of function series defined by ∞ j=0 f (t − jd), where f : R → R + ∪ {0} with f (t) = 0 for t < 0. It also discusses some biological applications that can be derived from these results, by considering each f (t − jd) as describing an isolated effect related to an application at time jd, and the sum of them as an accumulated effect.

2012

The paper gives a unified treatment of the summation of certain iterated series of the form ∑∞ ∑ ∞ n=1 m=1 an+m, where (an)n∈N is a sequence of real numbers. We prove that, under certain conditions, the double iterated series equals the difference of two single series. 1

Infinite-series representations find applications in many mathematical and engineering domains. The most common infinite-series representation is the power series. In this paper, we present a novel infinite-series representation of smooth functions and study its convergence. Additionally, we present applications, including an infinite-series representation of the gamma function.