combinatorics - How to prove a modified version of the binomial theorem?

The binomial identity:
$(x + y)^n = \sum_{k = 0}^{n}{n \choose k}x^{n - k}y^{k}$

seems closely related to:
$$(x + y)_n = \sum_{k = 0}^{n}{n \choose k}(x_{n - k})(y_{k})$$ For
$$(x)_{n} = (x) * (x- 1) * \dots * (x- n + 1), (x)_0 = 1$$
I'm trying to prove the second identity, for all $x \in \mathbb{C}$, so the standard induction proof does not apply. For positive $x, y$, a simple combinatorial argument should work, but I can't generalize to all $x \in \mathbb{C}$.

Answer

As in this answer, in general we have

$$\frac{d^n}{dt^n} f(t)g(t) = \sum_{k=0}^n \binom{n}{k} \left( \frac{d^k}{dt^k} f(t) \right) \left( \frac{d^{n-k}}{dt^{n-k}} g(t) \right).$$
(The proof is by induction on $n$, using the product rule and Pascal's identity.)

Now put $f(t) = t^y$ and $g(t) = t^x$. Then the two sides of the above equation are
$$\frac{d^n}{dt^n} t^y t^x = \frac{d^n}{dt^n} t^{x+y} = (x+y)_n t^{x+y-n},$$
and
$$\begin{align} \sum_{k=0}^n \binom{n}{k} \left( \frac{d^k}{dt^k} t^y \right) \left( \frac{d^{n-k}}{dt^{n-k}} t^x \right) &= \sum_{k=0}^n \binom{n}{k} (y)_k t^{y-n+k} \cdot (x)_{n-k} t^{x-k} \\ &= \sum_{k=0}^n \binom{n}{k} (x)_k (y)_{n-k} t^{x+y-n}. \end{align}$$
Putting $t = 1$ gives the desired result.

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Note that if we divide both sides of your equation by $n!$ then (recalling that $\binom{m}{n} = (m)_n/n!$) we end up with Vandermonde's identity
$$\binom{x+y}{n} = \sum_{k=0}^n \binom{x}{n-k} \binom{y}{k}.$$