Given that I have two formal power series:
$$ A(x) = \sum_{k \ge 0} a_k x^k $$
$$ B(x) = \sum_{k \ge 0} b_k x^k $$
The Cauchy Product gives a series
$$ C(x) = \sum_{k \ge 0} c_k x^k $$
$$ c_k = \sum_{n=0}^k a_n b_{k-n} $$
Which comes from taking the product of the two series $C(x)=A(x)B(x)$. What then, in terms of $A(x)$ and $B(x)$, is this series?
$$Y(x) = \sum_{k \ge 0} a_k b_k x^k $$
Best Answer
It is known as the Hadamard product $\rm\; f \star g\;$. There is no closed form for general hadamard products, but some classes of functions are known to be closed under such products, e.g. rational power series. However, algebraic series are not generally closed under Hadamard products, the standard example being $\rm\ f = g = (1-4x)^{-1/2} \;=\; \sum \binom{2n}{n}\ x^n.\ $ However, $\rm\: f\:$ rational, $\rm\: g\:$ algebraic $\rm\;\Rightarrow\; f\star g\;$ algebraic. Also D-finite power series are closed under Hadamard products, i.e. power series satisfying a linear differential equation with polynomial coefficients or, equivalently, series whose coefficients satisfy a linear recursive equation with polynomial coefficients.
Hadamard products can also be defined in terms of diagonals of multivariate series (and vice versa), e.g. see this paper, where you'll find some interesting connections with finite automata which, e.g. help one to easily observe that algebraic series over $\rm\mathbb{F}_p$ are closed under Hadamard product (which fails in characteristic 0)