Let $f(n)$ be defined on non-negative integers and define $S(m) = \sum_{n=0}^{m} \binom{m}{n} f(n)$. Depending on the choice of $f$, $S(m)$ may have a closed form; for example, when $f(n) = n$, then $S(m) = m 2^{m-1}$. My question is, does a general theory exist which relates the asymptotic behaviors of $f$ and $S$, and which does not rely on closed forms? Presumably the asymptotic behavior of $f$ is understood and that of $S$ is of interest.
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1$\begingroup$ Abel summation? $\endgroup$– Fan ZhengJan 9, 2017 at 3:00
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3$\begingroup$ Generating functions may help. If $\phi(x):=\sum_{n=0}^\infty f(n)x^n/n!$ then $e^x\phi(x):=\sum_{n=0}^\infty S(n)x^n/n!$. $\endgroup$– Pietro MajerJan 9, 2017 at 11:38
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