Multiplying two power series with negative indices

Question

Just wondering whether there is any special way (formula) to multiply to power series which contain negative indices, i.e. $\bigg(\sum\limits_{i=-m}^{\infty}a_ix^i\bigg)\bigg(\sum\limits_{j=-n}^{\infty}b_jx^j\bigg)$? Or the only way to do it is just by shifting indices? Cheers!

Answer 1

As a formal power series, the product is $\sum_{k \in \mathbb{Z}} c_kx^k$, where $$c_k = \sum_{i+j = k} a_ib_j$$ The trick is that since $a_i = 0$ for $i < -m$ and $b_j = 0$ for $j < -n$, the sum defining $c_k$ is actually finite. More precisely, it could me written as the finite sum $$c_k = \sum_{(i,j)\in S_k} a_ib_j \space\text{with}\space S_k = \{(i,j) \mid i+j =k, -m \leqslant i \leqslant k+n \space\text{and}\space -n \leqslant j \leqslant k+m\}$$