Изменения

Если <tex> r \in \mathbb{Z}_+ </tex>, <tex> \alpha k </tex> — мультииндекс, <tex> a </tex> - вектор, то <tex> (a_1 + ... + a_m)^r = \sum_{\alphak: (\alphak) = r} \frac{r!}{\alphak!} a^{\alphak} </tex>

<tex> \alpha k = (0, 0, \ldots, \overbrace{1}^{k}, 0, \ldots); a_k \cdot \frac{1!}{0!0! \ldots 1!0! ...} = 1 </tex>

<tex> (a_1 + ... + a_m)^{r + 1} = (a_1 + ... + a_m) \cdot \sum \frac{r!}{\alpha_1k_1! ... \alpha_mk_m!} \cdot a_1^{\alpha_k_{1}} ... a_m^{\alpha_k_{m}} = </tex>

<tex> = \sum \frac{r!}{\alpha_1k_1! ... \alpha_mk_m!} \cdot a_1^{\alpha_k_{1}+1} ... a_m^{\alpha_k_{m}} + \sum \frac{r!}{\alpha_1k_1! ... \alpha_mk_m!} \cdot a_1^{\alpha_k_{1}} a_2^{\alpha_2 k_2 + 1} ... a_m^{\alpha_k_{m}} + </tex><tex> \sum \frac{r!}{\alpha_1k_1! ... \alpha_mk_m!} \cdot a_1^{\alpha_k_{1}} ... a_{m-1}^{\alpha_k_{m - 1}} a_m^{\alpha_k_{m + 1}} = </tex>

<tex> = \sum_{\beta : |\beta| = r + 1; \beta_1 \ge 1} \frac{r! \beta_1}{\beta_1!\beta_2!...\beta_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} + \sum_{\beta : |\beta| = r + 1; \beta_2 \ge 1} \frac{r! \beta_2}{\beta_1!\beta_2!...\beta_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} + </tex> <ещё <tex> m - \alpha k </tex> суммы> = <tex> \sum_{|b| = r + 1} \frac{r! (b_1 + ... + b_m)}{b_1! ... b_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} </tex>;

<tex> (\alpha_1 k_1 + 1, \alpha_2 k_2 ... \alpha_mk_m) \to (\beta_1 ... \beta_m) </tex>

<tex> \sum_{(\alpha_1k_1...\alpha_mk_m); \alpha_i k_i \ge 0; \alpha_1 k_1 + ... + \alpha_m k_m = r} \frac{r!}{\alpha_1k_1! ... \alpha_mk_m!} \cdot a_1^{\alpha_k_{1}} ... a_m^{\alpha_k_{m}} = </tex><tex> \sum_{i_1 = 1}^m \sum_{i_2 = 1}^m ... \sum_{i_r = 1}^m a_{i_1} a_{i_2} ... a_{i_r} </tex>

<tex> m = 2; \alpha_1k_1, \alpha_2 k_2 = r - \alpha_1 k_1 </tex>

<tex> \sum_{\alpha_1 k_1 = 0}^{r} \frac{r!}{\alpha_1k_1!(r - \alpha_1k_1)!} \cdot a_1^{\alpha_1k_1} a_2^{r - \alpha_1k_1} = (a_1 + a_2)^r </tex>