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$(n D+1)$ Padé approximation

To generalize the concept of Padé approximation for functions to more than one variable, several definitions have been introduced. All definitions have advantages and disadvantages. The advantages of these approaches has been discussed in many articles. One of the main disadvantages of these methods are low convergence rate and the loss of information in computing with low degrees. In this work we present a new definition of the multivariate Padé approximation, treated the more than one variable to the coefficients. Our definition is designed to avoid such disadvantages with retaining the advantages of simple and easily computation. The main result obtained as consequence of this definition is some properties and convergence results of the Montessus De Ballore theorem.