A New Extended Padé Approximation and Its Application

We extend ordinary Padé approximation, which is based on a set of standard polynomials as , to a new extended Padé approximation (Müntz Padé approximation), based on the general basic function set ？？ (the particular case of Müntz polynomials) using general Taylor series (based on fractional calculus) with error convergency. The importance of this extension is that the ordinary Padé approximation is a particular case of our extended Padé approximation. Also the parameterization ( is the corresponding parameter) of new extended Padé approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research. 1. Introduction Rational approximations of an arbitrary function are an important topic in numerical analysis due to their high applications in physical sciences, chemistry, engineering, and other applied sciences [1, 2]. The Padé approximation is a particular and classical type of rational fraction approximation. The idea of this approximation is to expand a function as a ratio of two power series and determining both the numerator and denominator coefficients using the coefficients of Taylor series expansion of a function [1]. The Padé approximation is the best approximation of a function by a rational function of a given order [1]. The technique was developed around 1890 by Henri Padé, but it goes back to George Freobenius who introduced the idea and investigated the features of rational approximations of power series. The Padé approximation is usually superior when functions contain poles, because the use of rational function allows them to be well represented [1]. The Padé approximation often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge [1]. For these reasons, Padé approximation is used extensively in computer calculations. The Padé approximation has also been used as an auxiliary function in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods in some sense inspired by the Padé theory typically replace them. Since it provides an approximation to the function throughout the whole complex plane, the study of Padé approximation is simultaneously a topic in mathematical approximation theory and analytic function theory. The generalized Padé approximation is given in [2]. For the connection of Padé approximation with continued fractions and orthogonal polynomials, see [2]. Also multivariate Padé approximation was done by [3]. Two versions of this