We define the "standard" q-exponential as follows
$$ e_q(x) = 1 + \frac{1}{1} x+ \frac{1}{(1+q)} x^2 + \frac{1}{(1+q)(1+q+q^2)}x^3 ... =$$
$$ \sum_{i = 0}^{\infty} \left[ (1-q)^ix^i \prod_{j=1}^{i} \frac{1}{(1-q^j)} \right]$$
What I'm interested in studying is $e_q(x + y)$ inspired by the fact that
$$ e^{(x+y)} = e^{x} e^{y} $$
Is there a known "q-analog" of multiplication this way? What sort of identites exist of
$$ e_q(x+y) ?$$