I'm pretty sure that the sequences like $F_n=\sum_{k=1}^n \frac 1k$ are not traces of elementary functions on positive integers (take any reasonable definition of "elementary" you want, just make sure that all high school formulae are there). However, all proofs of non-elementarity I know make heavy use of differential fields and I do not see what and how to differentiate in this discrete setting. Any ideas, suggestions, or references?

P.S. I posted it on AoPS as well but then decided that there may be a slightly better chance to get an answer here :)