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Andrey Rekalo
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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 $f(k/n)$.

Andrey Rekalo
  • 22.3k
  • 12
  • 89
  • 122