How this operation is called?

Solution 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.)

Solution 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:
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Solution 3:

I came across this same binomial convolution in the following curious setting: consider the shift operator $S(a_n) = (a_{n+1})$ which maps $(a_0, a_1, a_2, \ldots) \mapsto (a_1, a_2, a_3, \ldots)$.

It is easy to check that $S$ is a derivation of this convolution, that is: $$ S ((a_n) \star (b_n)) = S (a_n) \star (b_n) + (a_n) \star S(b_n) $$ just by using Pascal's rule $ {n+1 \choose k} = {n \choose k} + {n+1 \choose k-1} $.

This can be used to give a proof of the form of the general solution to a linear recurrence (homogeneous, with constant coefficients): Just repeat the same linear algebra one does to give a proof of the form of the general solution to a linear homogeneous differential equation with constant coefficients, exchanging the derivative operator D, functions and the exponential functions by the shift operator S, sequences and the geometrical sequences.

I have not seen this used elsewhere and I do not know if it has other applications other then the one sketched above.