Consider the bivariate function with Taylor series development
around the origin. We know that a solution of the univariate Padé approximation problem for
is given by
Let us now multiply the j th row in and by () and afterwards divide the j th column in and by (). This results in a multiplication of numerator and denominator by . Having done so, we get
This quotient of determinants can also immediately be written down for a bivariate function . The sum will be replaced by the k th partial sum of the Taylor series development of and the expression by an expression that contains all the terms of degree k in . Here a bivariate term is said to be of degree .
If we define
then it is easy to see that and are of the form
We know that and are called Padé equations [3, 14]. So, the multivariate Padé approximant of order for is defined as