The (infinite) q-Pochhammer symbols are defined as
$$(a;q)_{\infty} := \prod_{k=0}^{\infty}(1-aq^{k})$$
I am interested a natural generalization of this symbol to multiple q's. In specific the 2 D and 3D cases which are defined as,
For 2 symbols q,r
$$(a;q,r)_{\infty} := \prod_{k=0}^{\infty}\prod_{l=0}^{\infty}(1-aq^{k}r^{l})$$
For 3 symbols q,r,s
$$(a;q,r,s)_{\infty} := \prod_{k=0}^{\infty}\prod_{l=0}^{\infty}\prod_{m=0}^{\infty}(1-aq^{k}r^{l}s^{m})$$
We can assume that $$|q|<1,|r|<1,|s|<1$$ for cases of interest.
I searched if they are discussed in the literature, but I could not find any leads. These generalizations naturally appear in the Grand Canonical partition function for fermions in a harmonic trap