Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inheri...

Kronrod's procedure is a method for estimating the error in Gaussian quadrature methods. Padé approximants are formal Gaussian quadrature formulas. In a previous paper, Kronrod's method was used to obtain .estimates of the error in Padé approximation. Using a new interpretation of this procedure and three different expressions of the error of Padé approximants, extensions of the method are obta...

A method is suggested for interpolating between small-variable and large-variable asymptotic expansions. The method is based on self-similar approximation theory resulting in self-similar root approximants. The latter are more general than the twosided Padé approximants and modified Padé approximants, including these as particular cases. Being more general, the self-similar root approximants gu...

Commonly used techniques to study non-perturbative aspects of the strong interactions have a deep connection with rational approximants, and in particular with Padé approximants to meromorphic functions. However, only recently this connection has been acknowledged and efforts at fully exploiting it are only starting. In this article I will briefly review the most prominent techniques used in no...

In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function f , defined on [−π, π], discontinuous at 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants of the system of functions (1, g1(z), g2(z)), where g1(z) = log(1 − z) and g2(z) is analytic, such that Re (g2(e)) = f(t). Conv...

The paper reviews the relation between Padé-type approximants of a power series and interpolatory quadrature formulas with free nodes, and that between Padé approximants and Gaussian quadrature methods. Quadrature methods are well-known. They are used for obtaining an approximate value of a definite integral, and are described in any book of numerical analysis. In this talk, we will show that P...

We obtain a result on convergence of Padé approximants for meromorphic functions of Stieltjes type. Connection of Padé approximants with Stieltjes moment problem plays a key role in the proof.

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions Fi, i = 1, . . . , 4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the Fi onto x = 0 or y = 0, namely, the Gauss function 2F1(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2...

We present a new set of algorithms for computation of matrix rational interpolants and one-sided matrix greatest common divisors. Examples of these interpolants include Padé approximants, Newton–Padé, Hermite–Padé, and simultaneous Padé approximants, and more generally M-Padé approximants along with their matrix generalizations. The algorithms are fast and compute all solutions to a given probl...