For motivation, consider multiplying a polynomial by the geometric series:
$$ \frac{P}{1-x} = \sum a_i x^i \sum x^j = \sum c_kx^k$$
By the cauchy product rule,
$$ c_k = \sum_{i=0}^k a_i$$
Now, we find that the coefficents of this new polynomial are the sum of first $k$ terms, but what I wish to know is there a function $Q$ such that:
$$ QP = \sum c_k a^k$$
WIth the property that
$$ c_k = \prod_{i=0}^k a_i$$
That is it is the product of first $k$ terms?