備忘録 -シグマの計算-

$$\begin{array}{rcl} \displaystyle \qty(\sum_{i=0}^m a_i\frac{s^i}{i!})^2 &=& \displaystyle \sum_{i=0}^m a_i\frac{s^i}{i!}\qty(\sum_{j=0}^m a_j\frac{s^j}{j!})\\ &=&\displaystyle \sum_{i=0}^m \sum_{j=0}^m a_i a_j\frac{s^{i+j}}{i!j!}\\ &=&\displaystyle \sum_{k=0}^{2m} \sum_{0\leq i,j \leq m}^{i+j=k} a_i a_j\frac{s^{k}}{i!j!}\\ &=&\displaystyle \sum_{k=0}^{2m} \qty(\sum_{0\leq i,j \leq m}^{i+j=k}\binom{k}{i} a_i a_j)\frac{s^{k}}{k!}\\ \end{array}$$

$$\begin{array}{rcl} \displaystyle \qty(\sum_{i=0}^m a_i\frac{s^i}{i!})^2 &=& \displaystyle \sum_{k=0}^{2m} \qty(\sum_{0\leq i,j \leq m}^{i+j=k}\binom{k}{i} a_i a_j)\frac{s^{k}}{k!}\\ &=& \displaystyle \sum_{k=0}^{m} \qty(\sum_{0\leq i+j=k\leq m} \binom{k}{i} a_i a_j )\frac{s^{k}}{k!} + \sum_{k=m+1}^{2m}\qty(\sum_{m+1\leq i+j=k\leq 2m} \binom{k}{i} a_i a_j)\frac{s^{k}}{k!}\\ &=& \displaystyle \sum_{k=0}^{m} \qty(\sum_{i=0}^k \binom{k}{i} a_i a_{k-i} )\frac{s^{k}}{k!} + O^{m+1}(s)\\ \end{array}$$