[MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n))

[JmsNxn]

I agree with james.

The residue at oo is linked to those at 0.

However in case our function is more complicated or we do not have a reflection formula ...
Im not sure.

Maybe something like

f(x) + h(f(1/x)) = g(x)

Can be used and considered a reflection kinda.

regards

tommy1729

Hey, Tommy;

I sent Caleb the PDF, not you yet; but "reflection formula" is more a term I'm using loosely. It's not literally \(f(x) = f(1/x)\) or something like this. It's more so we can use \(f(x)\) to construct \(f(1/x)\) and vice versa. Which is why I refer to it as a reflection formula. Essentially; we can reconstruct \(f(x)\) about \(x =0\) using the taylor coefficients of \(f(x)\) about \(x =\infty\)--I just like using "reflection formula" as a term; because it relates to Lambert series; and generalizes the simple reflection formula you see. I'm not actually just saying \(f(1/x) = h(f(x))\) or something!