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In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution:

$$F(n) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k)$$

This expected value operation seems to have a lot of nice properties with respect to differentiation (for instance, in this economics paper (Sah 1991) where the author proved some nice properties of these functions to deduce other things). So I suspect that this must be a named and well-studied phenomenon in the combinatorics/probability/convex optimization theory literature. But I couldn't find any discussion of it in the places I looked. (I tried "binomial distribution transformation", "binomial distribution transform", "expected value over binomial distribution", "expected utility function over binomial distribution", "convolution with binomial distribution", etc., but all the results I got were in applied economics/statistics).

Any ideas for where or under what name this might have been studied?

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    $\begingroup$ What exactly do you mean by "this"? The properties in the article you mentioned are just some auxiliary identities and inequalities that particular author needed for his model and none of them is anything an average mathematician would have any difficulty proving if he had needed them for anything. There is little point in trying to figure out all such properties of $F(n)$, much less in publishing them, so I doubt it has ever been done. On the other hand, if you need something particular and have trouble proving it, just post your question and we'll see what we can do. $\endgroup$
    – fedja
    Commented Apr 19, 2010 at 17:26

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This is a generalization of the binomial transform of the function $f(k)$. See, for instance, the Wikipedia article on binomial transform, and, in particular, the generalizations given therein. The Prodinger reference deals specifically with your expression for $F(n)$. Or, if you rewrite it as $$F(n) = (1-p)^n \sum_{k=0}^n \binom{n}{k} \left(\frac{p}{1-p}\right)^k f(k),$$ then you having a scaled version of the rising $k$-binomial transform of $f(k)$ as described in my 2006 paper with Laura Steil. At any rate, it appears the term you want is "binomial transform," and there is a small literature on its properties.

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

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