I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series.
The Hadamard Product
The Hadamard product is defined on generating functions $f(x)$ and $f(y)$ as $f(z)$ below:
$f(x) = \sum_{i=0}^\infty{c_i x^i}$
$f(y) = \sum_{i=0}^\infty{d_i y^i}$
$f(z) = \sum_{i=0}^\infty{c_i \cdot d_i z^i}$
My Particular Interest
Now suppose that we work on the series above with only the first N terms as possibly nonzero, and the rest will be considered to be zero.
Take the coefficients of $f(x)$ to be the reciprocals of the first $N$ naturals, e.g.
$f(x) = \sum_{i=1}^N{\frac{1}{i} x^{i+1}}$
Now suppose that $f(y)$ is in the form of an expression, i.e.
$f(y) = \frac{1-(2y)^{N-1}}{1 - 2y}$ represents the first N terms of the form $(2y)^N$
or
$f(y) = \frac{1-(n+1)x^N+Nx^{N+1}}{(x-1)^2}$ represents the first N naturals in series
or some similar function...
My Question
Can we find a simple expression, such as a recursion or formula, to represent the "Hadamard product" of these new series?
We could simply write out the $N$ terms in the resulting function $f(z)$. However, I'm wondering if there is a more compact way of representing this. I'm particularly interested when $f(x)$ is the series of reciprocals mentioned above. However, I am interested in finding this result for almost any $f(y)$. Is there a way to do this?
I'd like to consider both the cases when coefficients of $f(y)$ take on real values and when it is complex-valued.
