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This is the well-known theory of Puiseux series (substitute $T=1/n$ to match the notation in that website). The convergence of the series is mentioned there too. So in your dissertation, I would rec …

Here is a proof of the conjecture, a proof that also shows how to compute the integrals explicitly. The proof is somewhat similar to David Speyer's approach, but instead of using multivariable residu …

Andrew's comments showed me that in my first answer I was misunderstanding several aspects of his question. Since I am still not entirely sure that I am capturing the spirit of the problem, let be be …

The answer is that everything can be recovered, if $f$ is chosen suitably! There are $2^{\aleph_0}$ possibilities for $c$, so we can fix an injection $c \mapsto u_c$ from the set of possible $c$ into …

Yes, there exist nonlinear solutions. Multiplying by $e^{x+1}$ and setting $g(x):=e^x f(x)$ transforms the question into finding a solution to $g(x+1)=eg'(x)$ not of the form $e^x(ax+b)$. Start with …