**Notation**

A function *f*(*z*) is represented by a formal power series:

where *c*_{0} ≠ 0, by convention. The (*m*, *n*)th entry *R _{m, n}* in the Padé table for

*f*(

*z*) is then given by

where *P _{m}*(

*z*) and

*Q*(

_{n}*z*) are polynomials of degrees not more than

*m*and

*n*, respectively. The coefficients {

*a*} and {

_{i}*b*} can always be found by considering the expression

_{i}and equating coefficients of like powers of *z* up through *m* + *n*. For the coefficients of powers *m* + 1 to *m* + *n*, the right hand side is 0 and the resulting system of linear equations contains a homogeneous system of *n* equations in the *n* + 1 unknowns *b _{i}*, and so admits of infinitely many solutions each of which determines a possible

*Q*.

_{n}*P*is then easily found by equating the first

_{m}*m*coefficients of the equation above. However, it can be shown that, due to cancellation, the generated rational functions

*R*are all the same, so that the (

_{m, n}*m*,

*n*)th entry in the Padé table is unique. Alternatively, we may require that

*b*

_{0}= 1, thus putting the table in a standard form.

Although the entries in the Padé table can always be generated by solving this system of equations, that approach is computationally expensive. More efficient methods have been devised, including the epsilon algorithm.

Read more about this topic: Padé Table