📂Complex Anaylsis

Generalized Binomial Coefficients for Complex Numbers

Definition

For a complex number $\alpha \in \mathbb{C}$, the following is called a Binomial Coefficient. $$\binom{\alpha}{k} := \begin{cases} \displaystyle {{ \alpha ( \alpha - 1 ) \cdots ( \alpha - k + 1 ) } \over { k! }} & , k \in \mathbb{N} \\ 1 & ,k=0 \end{cases}$$

Description

Originally, binomial coefficients have an intuitive meaning only when $\alpha \in \mathbb{N}$, but if we consider just the calculation process, there is no need for it to be natural numbers. It can be extended to negative integers, real numbers, and even complex numbers.

Theorem

$$\sum_{j=0}^{k} \binom{\alpha}{k-j} \binom{\beta}{j} = \binom{\alpha + \beta}{k}$$

Proof

Strategy: Nothing but mathematical induction and tedious calculations.

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If $k = 0$ then $$1 \cdot 1 = 1$$ If $k = 1$ then $$\alpha \cdot 1 + 1 \cdot \beta = \alpha + \beta$$ Assuming that when $k \ge 1$, $\displaystyle \sum_{j=0}^{k} \binom{\alpha}{k-j} \binom{\beta}{j} = \binom{\alpha + \beta}{k}$ holds, $$\begin{align*} \binom{ \alpha + \beta }{ k + 1 } =& \binom{ \alpha + \beta }{ k } {{ \alpha + \beta - k } \over { k + 1 }} \\ =& \sum_{j=0}^{k} \binom{ \alpha }{ k - j } \binom{ \beta }{ j } \left( {{ \alpha - k + j } \over { k + 1 }} + {{ \beta - j } \over { k + 1 }} \right) \\ =& \sum_{j=0}^{k} \left[ {{ k - j + 1 } \over { k + 1 }} \binom{ \alpha }{ k - j + 1 } \binom{ \beta }{ j } + {{ j + 1 } \over { k + 1 }} \binom{ \alpha }{ k - j } \binom{ \beta }{ j + 1 } \right] \\ =& \binom{ \alpha }{ k + 1 } + \sum_{j=1}^{k} \left( {{ k - j + 1 } \over { k + 1 }} + {{ j } \over { k + 1 }} \right) \binom{ \alpha }{ k - j + 1 } \binom{ \beta }{ j } + \binom{ \beta }{ k + 1 } \\ =& \binom{ \alpha }{ k + 1 } + \sum_{j=1}^{k} \binom{ \alpha }{ k - j + 1 } \binom{ \beta }{ j } + \binom{ \beta }{ k + 1 } \\ =& \sum_{j=0}^{k+1} \binom{ \alpha }{ (k + 1) - j } \binom{ \beta }{ j } \end{align*}$$

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