I am trying to find a general expression for the coefficients of the power series that results from the multiplication of a polynomial and a power series. I have looked at this post Convolution and multiplication of polynomials is the same? which deals with multiplication of two finite-order polynomials, but this doesn't work for the case I have described, and I can't see how to adapt it.
In short, I have:
\begin{equation} f\left(x\right)=\sum_{i=0}^{n-1} \theta_i x^i \quad \mbox{and} \quad g\left(x\right)=\sum_{j=0}^\infty c_jx^j \quad \mbox{so} \quad f\left(x\right)g\left(x\right) = \sum_{j=0}^\infty\sum_{i=0}^{n-1} c_j\theta_i x^j x^i \end{equation}
so polynomial $f$ is of order $n-1$ and $g$ is of infinite order. However, I need this to have a single power of $x$ and can't see how to simplify this - any help gratefully received!