Are there any identities, papers/studies, posts, etc that go over
$$(\ln\zeta_n^k;q)_{\infty} = \prod_{m=0}^{\infty}(1-\frac{2\pi i k q^m}{n})$$ which is sometimes called the $q$-Pochhammer or quantum dilogarithm, but evaluated at $z = \ln\zeta_n^k$, i.e $n$-th root of unity to some power $k \leq n$. The motivation behind this is that this would be like a $q$-root of unity, as the quantum dilogarithm is also the $q$-exponential function, and hence $$e_q\left(\frac{2\pi i k}{n}\right) = \left(\frac{2\pi i k}{n};q\right)_{\infty}$$