So that the identities themselves do not become buried on an obscure page, we summarize them immediately:
$$
(x + y)^n = \sum\limits_k {\left( {\begin{array}{*{20}c}
n \\
k \\
\end{array} } \right)} x^k y^{n - k} , \begin{array}{*{20}c}
{integer n} \\
{or n real and |x/y| < 1} \\
\end{array}
$$
(1.1)
$$
\left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{r - 1} \\
k \\
\end{array} } \right) + \left( {\begin{array}{*{20}c}
{r - 1} \\
{k - 1} \\
\end{array} } \right), \begin{array}{*{20}c}
{real r} \\
{interger k} \\
\end{array}
$$
(1.2)
$$
\left( {\begin{array}{*{20}c}
n \\
k \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
n \\
{n - k} \\
\end{array} } \right), \begin{array}{*{20}c}
{integer n \geqslant 0} \\
{integer k} \\
\end{array}
$$
(1.3)
$$
\left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right) = \frac{r}
{k}\left( {\begin{array}{*{20}c}
{r - 1} \\
{k - 1} \\
\end{array} } \right), \begin{array}{*{20}c}
{real r} \\
{integer k \ne 0} \\
\end{array}
$$
(1.4)
$$
\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c}
{r + k} \\
k \\
\end{array} } \right)} = \left( {\begin{array}{*{20}c}
{r + n + 1} \\
n \\
\end{array} } \right), \begin{array}{*{20}c}
{real r} \\
{integer n \geqslant 0} \\
\end{array}
$$
(1.5)
$$
\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c}
k \\
m \\
\end{array} } \right)} = \left( {\begin{array}{*{20}c}
{n + 1} \\
{m + 1} \\
\end{array} } \right), integer m,n \geqslant 0
$$
(1.6)
$$
\left( {\begin{array}{*{20}c}
{ - r} \\
k \\
\end{array} } \right) = ( - 1)^k \left( {\begin{array}{*{20}c}
{r + k - 1} \\
k \\
\end{array} } \right), \begin{array}{*{20}c}
{real r} \\
{integer k} \\
\end{array}
$$
(1.7)
$$
\left( {\begin{array}{*{20}c}
r \\
m \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
m \\
k \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{r - k} \\
{m - k} \\
\end{array} } \right), \begin{array}{*{20}c}
{real r} \\
{integer m,k} \\
\end{array}
$$
(1.8)
$$
\sum\limits_k {\left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right)} \left( {\begin{array}{*{20}c}
s \\
{n - k} \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{r + s} \\
n \\
\end{array} } \right), \begin{array}{*{20}c}
{real r,s} \\
{integer n} \\
\end{array}
$$
(1.9)
$$
\sum\limits_k {\left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right)} \left( {\begin{array}{*{20}c}
s \\
{n + k} \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{r + s} \\
{r + n} \\
\end{array} } \right), \begin{array}{*{20}c}
{integer n,real s} \\
{integer r \geqslant 0} \\
\end{array}
$$
(1.10)
$$
\sum\limits_k {\left( {\begin{array}{*{20}c}
r \\
k \\
\end{array} } \right)} \left( {\begin{array}{*{20}c}
{s + k} \\
n \\
\end{array} } \right)( - 1)^k = ( - 1)^r \left( {\begin{array}{*{20}c}
s \\
{n - r} \\
\end{array} } \right), \begin{array}{*{20}c}
{integer n,real s} \\
{integer r \geqslant 0} \\
\end{array}
$$
(1.11)
$$
\sum\limits_{k = 0}^r {\left( {\begin{array}{*{20}c}
{r - k} \\
m \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{s + k} \\
n \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{r + s + 1} \\
{m + n + 1} \\
\end{array} } \right), \begin{array}{*{20}c}
{integer m,n,r,s \geqslant 0} \\
{n \geqslant s} \\
\end{array} }
$$
(1.12)
Parameters called real here may also be complex.