generalized binomial theorem

Statement

For any $a, y \in C$ , we have

(1 + y)^{a} = (\binom{a}{0}) + (\binom{a}{1}) y^{1} + (\binom{a}{2}) y^{2} + \dots

where the binomial coefficient is defined for all $k \in Z_{\geq 0}$

(\binom{a}{k}) = \frac{a (a - 1) (a - 2) \dots (a - k + 1)}{k!}

Look at the Taylor series expansion of $(1 + y)^{a}$ at $y = 0$ .