I found an old question asking for a proof that the factorial function is nonelementary, and the Claim 2 section of the answer there (by Vincenzo Oliva) doesn't quite make sense to me: https://math.stackexchange.com/a/1394130/314780.
My issue is when they claim that the $2n$-th Bernoulli number $B_{2n}$ is an elementary function of $n$. Their justification is that it is defined as the constant term of the $2n$-th Bernoulli polynomial $B_{2n}(m)$, and polynomials are elementary. But the fact that $B_2(m), B_4(m), B_6(m), ...$ are each individually elementary functions of $m$ does not imply that the function $B_{2n}(0)$ is an elementary function of $n$. Is there a way to fix this proof (or am I just misunderstanding something)?