There are several identities which resemble the binomial theorem. For starters, we have the binomial theorem itself: $$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}$$
But I just learned from the book "Concrete Mathematics", Exercise 5.37, that the "falling factorial" $x^{\underline{k}} = x(x-1)\ldots(x-k+1)$ satisfies a similar identity: $$(x+y)^\underline{n} = \sum_{k=0}^n \binom{n}{k} x^\underline{k} y^\underline{n-k}$$ The "rising factorial" $x^{\overline{k}} = x(x+1)\ldots(x+k-1)$ also satisfies such an identity.
Sometimes, the identity involves a product instead of a sum on the left side. If $f$ and $g$ are $n$-times differentiable functions on $\mathbb{R}$, then this generalization of the product rule holds: $$(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(k)} g^{(n-k)}$$ where $f^{(k)}$ denotes the $k$-th derivative of $f$, and the 0th derivative of a function is the function itself.
Question: Are there any more of these binomial-theorem like identities in other contexts? Are these identities part of some more general result, where we can axiomatize some conditions under which some "iterative" process satisfies a binomial-like theorem?
Best Answer
Some keywords you'll want to look into: binomial type, Appell sequence, Sheffer sequence, umbral calculus. The references in the corresponding Wikipedia articles are good too.
Edit: In some sense, all of these identities can be deduced from the last one. Setting $$f(t) = e^{xt}, g(t) = e^{yt}$$
produces the binomial theorem, and setting $$f(t) = (1 + t)^x = \exp (x \log (1 + t)), g(t) = (1 + t)^y = \exp ( y \log (1 + t))$$
produces the second identity. From this perspective one can think of the study of generalized binomial theorems as being all about generating functions of the form $\exp (x h(t))$ where $h(0) = 0$; setting $$f(t) = \exp (x h(t)), g(t) = \exp (y h(t))$$
produces a fairly general binomial theorem, especially if one writes $h(t) = \sum_{n \ge 1} h_n t^n$ as a formal power series in formal variables.