Capelli's Sum

Theorem

 * $\ds \paren {x + y}^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$

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 $\map P n$ be the proposition:
 * $\ds \paren {x + y}^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

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

So this is the induction hypothesis:
 * $\ds \paren {x + y}^{\overline m} = \sum_k \binom m k x^{\overline k} y^{\overline {m - k} }$

from which it is to be shown that:
 * $\ds \paren {x + y}^{\overline {m + 1} } = \sum_k \binom {m + 1} k x^{\overline k} y^{\overline {m + 1 - k} }$

Induction Step
This is the induction step:

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

Therefore:
 * $\ds \forall n \in \Z_{> 0}: \paren {x + y}^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$