Double q-analog of Pochhammer

Question

Has the function $$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$ been studied in the math literature? For example, does it obey any difference equation similar to the quantum dilogarithm, or have some modular properties? The only place I see it, is the theory of elliptic Gamma function.

One reference I found is this paper about integrable systems in physics: Mironov, Morozov, and Zenkevich - Duality in elliptic Ruijsenaars system and elliptic symmetric functions.

Answer 1

This product appears in permutation enumeration. See, for example, D. P. Roselle, Coefficients associated with the expansion of certain products (DOI), Proc. Amer. Math. Soc. 45 (1974), 144-150.