This should have been a comment but I don't have enough reputation points to post comments. 

The expression for $F(n)$ looks very similar to *the Bernstein approximation* (or <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>) of the function $f(.)$. Actually, it would be the Bernstein polynomial with respect to $p$ if the values $f(k)$ are replaced with $f(k/n)$.