上指标求和公式练习

\[\begin{aligned} \sum_{i=l_1}^{r_1}\sum_{j=l_2}^{r_2}\binom{i+j}{i}&=\sum_{j=l_2}^{r_2}\sum_{i=l_1}^{r_1}\binom{i+j}{i} \\ &=\sum_{j=l_2}^{r_2}(\binom{r_1+j+1}{r_1}-\binom{l_1+j}{l_1-1}) \\ &=\sum_{j=l_2}^{r_2}\binom{r_1+j+1}{r_1}-\sum_{j=l_2}^{r_2}\binom{l_1+j}{l_1-1} \\ &=\sum_{i=l_2+r_1+1}^{r_2+r_1+1}\binom{i}{r_1}-\sum_{i=l_2+l_1}^{r_2+l_1}\binom{i}{l_1-1} \\ &=\binom{r_1+r_2+2}{r_1+1}-\binom{r_1+l_2+1}{r_1+1}+\binom{l_1+l_2}{l_1}-\binom{l_1+r_2+1}{l_1} \end{aligned} \]

\[\begin{aligned} \sum_{j={l_2}}^{r_2}\sum_{i={l_1}}^{r_1}\binom{i+j}{j}&=\sum_{j={l_2}}^{r_2}\sum_{i=l_1+j}^{r_1+j}\binom{i}{j}\\ &=\sum_{j={l_2}}^{r_2}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})\\ &=\sum_{j=0}^{r_2}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})-\sum_{j=0}^{l_2-1}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})\\ &=\sum_{j=0}^{r_2}\binom{r_1+j+1}{j+1}-\sum_{j=0}^{r_2}\binom{l_1+j}{j+1}-\sum_{j=0}^{l_2-1}\binom{r_1+j+1}{j+1}+\sum_{j=0}^{l_2-1}\binom{l_1+j}{j+1}\\ &=\sum_{j=1}^{r_2+1}\binom{r_1+j}{j}-\sum_{j=1}^{r_2+1}\binom{l_1+j-1}{j}-\sum_{j=1}^{l_2}\binom{r_1+j}{j}+\sum_{j=1}^{l_2}\binom{l_1+j-1}{j}\\ &=\binom{r_1+r_2+2}{r_2+1}-\binom{l_1+r_2+1}{r_2+1}-\binom{r_1+l_2+1}{l_2}+\binom{l_1+l_2}{l_2} \end{aligned} \]

\[\begin{aligned} &\sum_{i=0}^n\binom{i}{m} =\sum_{i=m}^n\binom{i}{m} =\sum_{i=m}^n\binom{i}{m}+\sum_{i=0}^{k(k<m)}\binom{i}{m} =\binom{n+1}{m+1} \end{aligned} \]