Related papers: A novel advancement in the study of Appell polynom…
The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Pad\'e polynomials over the Pad\'e approximation procedure. The first part of the paper is devoted to…
Pad\'e approximants are rational functions whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different…
The $(u,v)$-Pad\'e approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and…
It is described how the Hermite-Pad\'e polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Pad\'e polynomials. A…
A polynomial family $\{p_n(x)\}$ is Appell if it is given by $\frac{e^{xt}}{g(t)} = \sum_{n=0}^\infty p_n(x)t^n$ or, equivalently, $p_n'(x) = p_{n-1}(x)$. If $g(t)$ is an entire function, $g(0)\neq 0$, with at least one zero, the…
In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…
The main purpose of this paper is to compare the convergence properties of Pad\'e approximants and rational Hermite-Pad\'e approximants for some model class of multivalued analytic functions based of the inverse Zhoukovsky transform. We…
En utilisant des approximants de Hermite-Pad\'e de fonctions exponentielles, ainsi que des d\'eterminants d'interpolation de Laurent, nous minorons la distance entre un nombre alg\'ebrique et l'exponentielle d'un nombre alg\'ebrique non…
We apply the technique of self-similar exponential approximants based on successive truncations of continued exponentials to reconstruct functional laws of the quasi-exponential class from the knowledge of only a few terms of their power…
Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in…
We present a novel method for calculating Pad\'e approximants that is capable of eliminating spurious poles placed at the point of development and of identifying and eliminating spurious poles created by precision limitations and/or noisy…
A survey of direct and inverse type results for row sequences of Pad\'e and Hermite-Pad\'e approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Pad\'e approximation when it is known that the sequence…
Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Pad\'{e} approximants or other rational functions constructed from…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
In this article, we construct new Pad\'{e} approximations for the \emph{product} of binomial functions and powers of logarithmic functions. While several explicit Pad\'{e} approximants are known for powers of exponential functions, binomial…
In this work, we use rational approximation to improve the accuracy of spectral solutions of differential equations. When working in the vicinity of solutions with singularities, spectral methods may fail their propagated spectral rate of…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
Given a system of functions $\textup{\textbf{F}}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common…
Pad\'e approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of…
We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Pad\'e approximants that are known to provide the best…