Newton's Generalization of the Binomial Theorem

# Newton's Generalization of the Binomial Theorem

Recall from The Binomial Theorem page that for all $x, y \in \mathbb{R}$ and for all $n \in \{0, 1, 2, ... \}$ we have that the expansion of the binomial $(x + y)^n$ is given by the formula:

(1)
\begin{align} \quad \binom{n}{0} x^ny^0 + \binom{n}{1} x^{n-1}y^1 + ... + \binom{n}{n-1} x^1y^{n-1} + \binom{n}{n} x^0y^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k \end{align}

The binomial coefficients are defined for for nonnegative integers $k$ where $0 \leq k \leq n$ to be $\binom{n}{k} = \frac{n!}{k! (n-k)!}$. The definition above works amazingly well for nonnegative integers $n$ but what if we would rather expand the the binomial $(x + y)^{\alpha}$ where $\alpha$ is any real number? Such a formula exists but it requires a little reworking of our definition of the binomial constants.

For $\alpha \in \mathbb{R}$ and $k \in \{ 0, 1, 2, ... \}$ we will define the binomial coefficient $\binom{\alpha}{k}$ as follows:

(2)
\begin{align} \quad \binom{\alpha}{k} = \left\{\begin{matrix} \frac{\alpha \cdot (\alpha-1) \cdot (\alpha-2) \cdot ... \cdot (\alpha - k + 1)}{k!} && \mathrm{if} \: k \geq 1\\ 0 && \mathrm{if} k = 0 \end{matrix}\right. \end{align}

Note that if $\alpha = n \in \{0, 1, 2, ... \}$ and $0 \leq k \leq n$ then $\binom{\alpha}{k} = \binom{n}{k} = \frac{n \cdot (n - 1) \cdot (n - 2) \cdot ... \cdot (n - k + 1)}{k!} = \frac{n!}{k!(n - k)!}$ which is our usual definition for the binomial constants.

We now ready to state, but not prove, Newton's generalization of the binomial theorem.

 Theorem 1 (Newton's Generalization of the Binomial Theorem): Let $x, y \in \mathbb{R}$ where $0 \leq \lvert x \rvert < \lvert y \rvert$ and let $\alpha \in \mathbb{R}$. Then the expansion of the binomial $(x + y)^{\alpha}$ is given by the infinite series $\displaystyle{(x + y)^{\alpha} = \binom{\alpha}{0} x^0y^{\alpha} + \binom{\alpha}{1} x^1 y^{\alpha - 1} + \binom{\alpha}{2} x^{\alpha} y^{2-\alpha} + ... = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k y^{\alpha - k}}$.

It is important to note that Newton's generalization of the binomial theorem results is an infinite series. Furthermore, one should note that if instead we have $0 \leq \lvert y \lvert < \lvert x \rvert$ then we can instead swap the variables $x$ and $y$ in the above formula.