Torelli's Sum

Theorem

 * $\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x \left({x - k z + 1}\right)^{\overline {k - 1} } \left({y + k z}\right)^{\overline {n - k} }$

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

Proof
From Rising Factorial as Factorial by Binomial Coefficient:


 * $\left({x + y}\right)^{\overline n} = n! \dbinom {x + y + n - 1} n$

Recall Sum over $k$ of $\dbinom {r - t k} k$ by $\dbinom {s - t \left({n - k}\right)} {n - k}$ by $\dfrac r {r - t k}$:


 * $\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \left({n - k}\right)} {n - k} \frac r {r - t k} = \binom {r + s - t n} n$

Let the following substitutions be made:
 * $r \gets x$
 * $t \gets -\left({1 - z}\right)$
 * $s \gets y - 1 + n z$

and so to obtain:


 * $\displaystyle \dbinom {x + y + n - 1} n = \sum_k \dbinom {x + \left({1 - z}\right) k} k \dbinom {y - 1 + n z + \left({n - k}\right) \left({1 - z}\right)} {n - k} \dfrac x {x + \left({1 - z}\right) k}$

Then:

and:

Hence: