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?