Product between an infinite and a finite power series

Question

Is there any demonstration or general formula of how the product between an infinite and a finite power series takes form?

Answer 1

The product of power series has this form, where $a$ and $b$ are coefficient sequences. $a$ has infinitely many terms and $b$ has $n$ terms.

$$\sum_{i=0}^\infty a_ix^i\ \cdot\sum_{j=0}^nb_jx^j$$

The series can be distributed term by term.

$$=\sum_{i=0}^\infty\sum_{j=0}^na_ib_jx^{i+j}$$

Now we have to group the like powers of $x$, which is equivalent to grouping the terms with the same value of $i+j$. Let $k=i+j$. for a particular value of $k$, we can take all the terms starting at $i=k,j=0$ and iterating down to either $i=0,j=k$ or $i=k-n,j=n$, whichever comes first. The sum can then be written as:

$$\sum_{k=0}^\infty\sum_{j=0}^{min(n,k)}a_{k-j}b_jx^k$$

We can create a new coefficient sequence $c$ such that:

$$c_k=\sum_{j=0}^{min(n,k)}a_{k-j}b_j$$

The product can then be written as:

$$\sum_{i=0}^\infty a_ix^i\ \cdot\sum_{j=0}^nb_jx^j=\sum_{k=0}^\infty c_kx^k$$

With more information about $a$ and $b$, it may be possible to simplify the expression for $c$.