Lesson 4

Let us now summarize some standard operations. In the formulas below, we use the convention \(a_{j}=0\) for \(j<0\). \[\begin{align*}\alpha \mathcal{A}(x) + \beta \mathcal{B}(x)&=\sum_{n}(\alpha a_{n}+\beta b_{n})x^{n}\\ x^{k}\mathcal{A}(x)&=\sum_{n} a_{n-k}x^{n}\\ \mathcal{A}(cx)&=\sum_{n}c^{n}a_{n}x^{n}\\ \mathcal{A}(x)\mathcal{B}(x)&=\sum_{n} \left(\sum_{k}a_{k}b_{n-k}\right)x^{n}\\ \frac{\mathcal{A}(x)-\sum_{i < k}a_ix^i}{x^k}&=\sum_{n}a_{n+k}x^{n}\\ \mathcal{A}'(x)&=\sum_{n} (n+1)a_{n+1}x^{n}\\ x\mathcal{A}'(x)&=\sum_{n} na_{n}x^{n}\\ \frac{1}{1-x}\mathcal{A}(x)&=\sum_{n}\left(\sum_{k \le n}a_{k}\right)x^{n}\end{align*}\]