Binomial Theorem/Extended

Theorem

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

Let $z \in \C$ be a complex number such that $\cmod z < 1$.

Then:

$\ds \paren {1 + z}^r = \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} z^{\alpha + k}$

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

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $42$ (Solution)