How can I prove this property (found in this wikipedia article) on the $\alpha$-factorial functions:
$$(\alpha n - 1)!_{(\alpha)} = \sum_{k=0}^{n-1}\binom{n-1}{k+1}(-1)^{k}\times\binom{\frac{1}{\alpha}+k-n}{k+1} \binom{\frac{1}{\alpha}-1}{k+1}\times\left(\alpha(k+1)-1\right)!_{(\alpha)}\left(\alpha(n-k-1)-1\right)!_{(\alpha)},$$ where $$n!_{(\alpha)} = \begin{cases}n (n-\alpha)!_{(\alpha)} & \text{if } n>\alpha;\\ n & \text{if } 1\leq n \leq \alpha;\\ (n+\alpha)!_{(\alpha)}/(n+\alpha) &\text{if } n\leq 0 \text{ and is not a negative multiple of } \alpha. \end{cases}$$
I tried to prove it by induction.
However, in base step, I start with $n = 1$. On the left hand side of the formula, it's equal to $\alpha-1$ by definition of multifactorial.
While, on the right hand side of the formula, it's equal to zero since $\binom{n-1}{k+1} = 0$ when $n = 1$ and $k = 0$.
It contradicts the base step.
Did I make something wrong?
Any help will be appreciated.