How this operation is called?

Question

This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients:

$$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k) $$

Particularly,

$$a^n\star b^n=(a+b)^n$$

following binomial theorem.

I just wonder if there is a name for such operation and where I can read about its properties.

Answer 1

It's called a binomial convolution in Graham, Knuth, and Patashnik's Concrete Mathematics. I don't have that text in front of me (but I bet someone here can give you a page number), but here's a reference on the Fermat's Last Theorem blog.

It would also be worth checking out Section 2.3 of Wilf's Generatingfunctionology. This is on exponential generating functions. The property of interest is that if $F(x)$ and $G(x)$ are the exponential generating functions of $f(n)$ and $g(n)$, respectively, then $F(x)G(x)$ is the exponential generating function of $f(n) \star g(n)$.

(FYI: You can download the second edition of Generatingfunctionology from Wilf's web site.)

Answer 2

Remark that one can employ the powerful umbral calculus to compute closed forms for many such binomial convolutions of special functions, e.g. from p. 161 of Roman: Umbral Calculus:

Answer 3

It corresponds to multiplying exponential generating functions (check generatingfunctionology).