A Riff on Padé approximants

Suppose I have a transcendental function w(z):\mathbb{C}\rightarrow\mathbb{C}, and I wish
to construct a new function from all Padé approximants to it, like thus:

\sum_{m=0}^{\infty} \sum_{m=0}^{\infty} (-1)^{m+n} R_{m,n}(z)

Why do this? Well, the idea here is to cancel out the common and make something new.

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