Related papers: Functional Power Series

Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We…

The original derivation of Power Functional Theory, Schmidt and Brader, JCP 138, 214101 (2013), is reworked in some detail with a view to clarifying and simplifying the logic and making explicit the various functional dependencies. We note…

We propose a new class of generative diffusion models, called functional diffusion. In contrast to previous work, functional diffusion works on samples that are represented by functions with a continuous domain. Functional diffusion can be…

In this work we have considered formal power series and partial differential equations, and their relationship with Coding Theory. We have obtained the nature of solutions for the partial differential equations for Cycle Poisson Case. The…

This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…

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…

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…

Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is…

We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en,…

The Bessel function of the first kind $J_{N}\left(kx\right)$ is expanded in a Fourier-Legendre series, as is the modified Bessel functions of the first kind $I_{N}\left(kx\right)$. The purpose of these expansions in Legendre polynomials was…

In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving…

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of…

We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…

We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.

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…

The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet…

This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…

The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…

In this article, a class of analytic functions is investigated and their some properties are established. Several recurrence relations and various classes of bilinear and bilateral generating functions for these analytic functions are also…

We discuss the functional lazy techniques in generation and handling of arbitrarily long sequences of derivatives of numerical expressions in one ``variable''; the domain to which the paper belongs is usually nicknamed ``Automatic…