Capelli's Sum

Theorem

 * $\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$

where:
 * $\dbinom n k$ denotes a binomial coefficient
 * $x^{\overline k}$ denotes $x$ to the $k$ rising.

Proof
The proof proceeds by induction on $n$.

For all $n \in \Z_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$

Basis for the Induction
$P \left({1}\right)$ is the case:

Thus $P \left({1}\right)$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({m}\right)$ is true, where $m \ge 1$, then it logically follows that $P \left({m + 1}\right)$ is true.

So this is the induction hypothesis:
 * $\displaystyle \left({x + y}\right)^{\overline m} = \sum_k \binom m k x^{\overline k} y^{\overline {m - k} }$

from which it is to be shown that:
 * $\displaystyle \left({x + y}\right)^{\overline {m + 1} } = \sum_k \binom {m + 1} k x^{\overline k} y^{\overline {m + 1 - k} }$

Induction Step
This is the induction step:

So $P \left({m}\right) \implies P \left({m + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \Z_{> 0}: \left({x + y}\right)^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$