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On Wikipedia it says that the $[m/n]_f$ Padé approximant to a function $f(x)$ is the rational function $$R(x)=\frac{a_0+a_1x+\cdots+a_m x^m}{1+b_1 x+\cdots+b_n x^n} $$ such that $$R^{(k)}(0)=f^{(k)}(0) $$ for all $0 \leq k \leq m+n$.

This definition makes me think that $[m+n/0]_f,[m+n-1/1]_f,\dots,[0/m+n]_f$ are all a "similar class" in terms of the amount of accuracy provided. My questions are: What sets them apart? Is there anything more significant than the number of zeros/poles? Which one is the best choice when trying to approximate a given function to degree $m+n$?

Thank you!

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    They are set apart by the fact that they are different functions, there is nothing more to it. As for the best choice, it can be any of them - that depends on the function you approximate. – 2017-01-17

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