Determinants involving q-Stirling numbers

Let S[i, j] denote the q-Stirling numbers of the second kind. We show that the determinant of the matrix (S[s + i + j, s + j])_0≤i,j≤n is given by the product q^{3(s+n+1)-3(s)}·[s]⁰ · [s + 1]¹ ⋯ [s + n]ⁿ. We give two proofs of this result, one bijective and one based upon factoring the matrix. We also prove an identity due to Cigler that expresses the Hankel determinant of q-exponential polynomials as a product. Lastly, a two variable version of a theorem of Sylvester and an application are presented.