TITLE: Why is there no formula for partial sums of some simple series?
QUESTION [14 upvotes]: 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 :)

REPLY [2 votes]: I am not sure what you mean by "traces", but there is certainly a theory of non-elementarity in the difference equation setting. One of the canonical early papers is:
MR1187234 (94a:39006) 
Petkovšek, Marko(SV-LJUB)
Hypergeometric solutions of linear recurrences with polynomial coefficients. (English summary) 
J. Symbolic Comput. 14 (1992), no. 2-3, 243–264. 
And there is also Wilf-Zeilberger and the book of Petkovsek and Zeilberger (A=B) which probably covers this in greater detail. The methods are still those of differential algebra.