Related papers: On Transforming the Generalized Exponential Power …
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…
This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…
A detailed analysis of the remainder obtained by truncating the Euler series up to the $n$th-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse…
We derive and discuss a technique for manipulating power series which is complementary to standard procedures. We begin with the translation operator, but we express the operator as an infinite product instead of expanding it as a series…
From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing…
This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
Efficient algorithms are known for many operations on truncated power series (multiplication, powering, exponential, ...). Composition is a more complex task. We isolate a large class of power series for which composition can be performed…
The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej\'{e}r's theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to…
Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…
In many physical problems, it is important to capture exponentially-small effects that lie beyond-all-orders of a typical asymptotic expansion; when collected, the full expansion is known as the trans-series. Applied exponential asymptotics…
Euler's formula, an extraordinary mathematical formula, establishes a vital link between complex-valued operations and trigonometric functions, finding widespread application in various fields. With the end of Moore's Law, electronic…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.
The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q-exponential functions by one more parameter. we…
In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…
We offer new Tauberian theorems for a generalized partition function as our main result. Our analysis provides insight into asymptotic behavior of power series with arithmetic functions as coefficients.
As was initially shown by Brent, exponentials of truncated power series can be computed using a constant number of polynomial multiplications. This note gives a relatively simple algorithm with a low constant factor.