Binomial Theorem/Extended

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Theorem

Let $r, \alpha \in \C$ be complex numbers.

Let $z \in \C$ be a complex number such that $\left|{z}\right| < 1$.


Then:

$\displaystyle \left({1 + z}\right)^r = \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} z^{\alpha + k}$

where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.


Proof


Sources