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What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

Some extended comments: Investigations of this type appear in the theory of the Mellin transform and, in particular, the use of Ramanujan's master heuristic (theorem, formula) as illustrated in the MO-...
• 8,551

What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

This is also just a comment but large: It might be worthwhile to try poking around with this formula on functions which cannot be analytically continued to see what arises. Trying to break the formula ...
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What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

First of all, to make sense of $f(-n)$ we need some assumptions about $f$. For example, let $$\sum_{n=0}^\infty f(n)z^n\quad\quad\quad\quad (1)$$ be a series with positive radius of convergence. Then ...
• 81.6k

What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

I'll convert my comment to an answer. One issue that you're implicitly sweeping under the rug is that we need to be able to make sense of the evaluation of $f$ at negative integers. Given an arbitrary ...
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• 81.6k

Holomorphic maps from a Riemann surface of infinite genus

Edit: as Moishe points out, my answer (just below) is for a different, and easier, question. I will leave the answer up, as it does feel “related”. The answer is no. I find it conceptually easier to ...
• 21.1k
Accepted

Quantitative analytic continuation estimate for functions small except on a small set

This conjecture is correct. Take $K=e$, and let $\gamma\leq 1/4$; we will fix $\gamma$ later. First we give a crude estimate of $c_0$. Let $g(z)=\sum_{1}^\infty c_nz^n.$ Since $|c_n|\leq e^n$, we ...
• 81.6k
Accepted

Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$ satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...
• 81.6k

Quantitative analytic continuation estimate for a function small on a set of positive measure

• 85.1k
Accepted

• 81.6k
Accepted

• 81.6k