• $$\large \sum_{j=1}^{n}a_j \quad 或 \quad \sum_{1 \leq j \leq n} a_j \tag{1}$$

• $$\large (\sum_{R(i)}a_j)(\sum_{S(j)}b_j) = \sum_{R(i)}(\sum_{S(j)}a_ib_j) \tag{4}$$

• $$\large \sum_{R(i)}\sum_{S(j)}a_{ij} = \sum_{S(j)}\sum_{R(i)}a_{ij} \tag{7}$$

• $$\large \sum_{R(i)}(b_i+c_i) = \sum_{R(i)}b_i + \sum_{R(i)}c_i \tag{8}$$

• $$\large \sum_{R(i)}\sum_{S(i,j)}a_{ij} = \sum_{S'(j)}\sum_{R'(i,j)}a_{ij} \tag{9}$$

• $$\large \sum_{i=1}^n\sum_{j=1}^ia_{ij} = \sum_{j=1}^n\sum_{i=j}^na_{ij} \tag{10}$$

• $$\large \sum_{i=0}^n\sum_{j=0}^ia_ia_j = \frac{1}{2}((\sum_{i=0}^na_i)^2+(\sum_{i=0}^na_i^2)) \tag{13}$$

• $$\large \sum_{0\leq j\leq n}ax^j = a(\frac{1-x^{n+1}}{1-x}) \tag{14}$$

• $$\large \sum_{0\leq j\leq n}(a+bj) = a(n+1)+\frac{1}{2}bn(n+1) \tag{15}$$

• $$\large \delta_{ij}=[i=j]=\begin{cases} 1, & 如果i=j\\0, & 如果i\neq j\end{cases} \tag{19}$$

• $$\large [R(i)][S(i,j)] = [R(i)且S(i,j)]= [S'(j)][R'(i,j)] \tag{21}$$