Adapted multivariate Padé approximation

To generalize the concept of Padé approximation for functions to more than one variable, several definitions have been introduced. We distinguish two types of definitions, the homogeneous multivariate Padé approximation and the general multivariate Padé approximation. Both definitions have advantages and disadvantages. In this work we present a new definition, of the multivariate Padé approximation, adapted to one class of functions. This definition is designed to avoid disadvantages of both definitions. The idea is that special cases deserve special treatment, which will enable approximants to show the character of function to approach and thus reduce the error of approximation and the computation cost. The main result obtained as consequence of this definition is some convergence results of multivariate Stieltjes series and a generalization of the Montessus De Ballore theorem for this class of multivariate functions.