Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential singularity. The generating function is then $$ g(x) = \sum_n s_n x^n $$, which seems not to simplify to a closed form expression for any $f$. For example $f(n)=1/n^3$ does not give a closed result it seems. Can anyone give a nontrivial counterexample for $f$ where the generating function gives a closed form? Are generating functions useful for these types of series?
2026-03-28 02:03:36.1774663416
Are generating functions ever analytic for logarithmic series?
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