2 Section 3, Id. 1: \(q\)-Stirling of the second kind
For all \(q : \mathbb {Q}\) (including \(q=0,1\)), \(m,n \ge 0\):
\[ F(m,n) \; =\; \sum _k (-1)^{n-k} \binom {k}{m}_q\, S_2^q(n,k)\, q^{n-k} \; =\; S_2^q(n+1,m+1). \]
Certificate and boundary: . Proved by direct double induction on \((m,n)\); no uniqueness theorem needed.