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 Bernstein polynomial) of the function $f(.)$. Actually, it would be the Bernstein polynomial with respect to $p$ if the values $f(k)$ are replaced with $f(k/n)$.

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 Bernstein polynomial) to the function $f(.)$. Actually, it would be the Bernstein polynomial with respect to $p$ if the values $f(k)$ are replaced with $f(k/n)$.