User:Cornince/no2

$$\begin{align} &\sum_{i=0}^{n} (-1)^{i} {n \choose i} z(n-i+(m+1)) - \sum_{i=0}^{n} (-1)^{i} {n \choose i} z(n-i+m) \\ &= z(n+1+m) - (-1)^{n} z(m) + \sum_{i=1}^{n} (-1)^{i} {n \choose i} z(n-i+m+1) - \sum_{i=0}^{n-1} (-1)^{i} {n \choose i} z(n-i+m) \\ &= z(n+1+m) + (-1)(-1)^{n} z(m) + \sum_{i=1}^{n} (-1)^{i} {n \choose i} z(n-i+m+1) + (-1) \sum_{i=1}^{n} (-1)^{(i-1)} {n \choose (i-1)} z(n-(i-1)+m) \\ &= z((n+1)+m) + (-1)^{(n+1)} z(m) + \sum_{i=1}^{n} (-1)^{i} {n \choose i} z(n-i+m+1) + \sum_{i=1}^{n} (-1)^{i} {n \choose i-1} z(n-i+m+1) \\ &= z((n+1)+m) + (-1)^{(n+1)} z(m) + \sum_{i=1}^{n} (-1)^{i} \left [ {n \choose i} + {n \choose i-1} \right ] z(n-i+m+1) \\ &= z((n+1)+m) + (-1)^{(n+1)} z(m) + \sum_{i=1}^{n} (-1)^{i} {(n+1) \choose i} z((n+1)-i+m) \\ &= \sum_{i=0}^{(n+1)} (-1)^{i} {(n+1) \choose i} z((n+1)-i+m) \end{align}\,$$

$$\sum_{k=0}^{n} (-1)^{k} {n \choose k} (n - k) ^ n = n! \,$$

$$n \ge 0$$

$$n \ge a \ge 0$$

$$\sum_{k=0}^{n} {n \choose k} x^k (n-k)^a = \frac{n!}{(n-a)!} (1+x)^{n-a}$$

$$a = 0\,$$

$$\sum_{k=0}^{n} {n \choose k} x^k (n-k)^0 = \frac{n!}{(n-0)!} (1+x)^{n-0}$$

$$\sum_{k=0}^{n} {n \choose k} x^k = (1+x)^n$$

$$a+1 \le n\,$$

$$\frac{\partial}{\partial x} \sum_{k=0}^{n} {n \choose k} x^k (n-k)^a = \frac{\partial}{\partial x} \frac{n!}{(n-a)!} (1+x)^{n-a}$$

$$\lim_{\Delta x \rightarrow 0} \cfrac{\sum_{k=0}^{n} {n \choose k} (x+\Delta x)^k (n-k)^a - \sum_{k=0}^{n} {n \choose k} x^k (n-k)^a}{\Delta x}\!\,$$

$$=\sum_{k=0}^{n} {n \choose k} (n-k)^a \lim_{\Delta x \rightarrow 0} \cfrac{(x + \Delta x)^k - x^k}{\Delta x}\!\,$$

$$=\sum_{k=0}^{n} {n \choose k} (n-k)^a \lim_{\Delta x \rightarrow 0} \cfrac{\sum_{j=0}^{k} {k \choose j} x^j \Delta x^{k-j} - \sum_{j=0}^{k} {k \choose j} x^j}{\Delta x}\!\,$$

$$=\sum_{k=0}^{n} {n \choose k} (n-k)^a \lim_{\Delta x \rightarrow 0} \cfrac{\sum_{j=0}^{k-1} {k \choose j} x^j \Delta x^{k-j}}{\Delta x}\!\,$$