Prove -1 Transformation (A binomial identity):$\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$

Question

Prove the following binomial identity :

$$\begin{align*} \binom{-p}{q}=\binom{p+q-1}{q}(-1)^q \end{align*} $$ I played with $\binom{-p}{q}$ for a while but found nothing .

Answer 1

Note that for real $\alpha$ and positive integer $q$, $\alpha\choose q$ is defined by: $${\alpha\choose q}=\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-(q-1))}{q!}.$$ So: $${-p\choose q}= \frac{(-p)(-p-1)(-p-2)\cdots(-p-(q-1))}{q!}= (-1)^q\cdot\frac{p(p+1)(p+2)\cdots (p+(q-1))}{q!}=\\ =(-1)^q\cdot\frac{(p+q-1)!}{q!(p-1)!}= (-1)^q{p+q-1\choose q}.$$