In my research, I have encountered the following equivalence identity:
For $a,b,n\in \mathbb{N}$, the equivalence $$ \left({\sum_{0\leq w < b}n^w}\right)^a \substack{\equiv\\n^b} \sum_{0\leq w < b}{a-1+w\choose w} n^w $$ holds.
One way of proving this equivalence relation is by induction on $b$; it seems to be elementary identity, and I am guessing that it has already been encountered.
Question: Does the above identity have an established name?
References welcome.