Poincaré series of the Weyl group of elliptic Lie algebras $A^{(1,1)}_1$ and $A^{(1,1) \ast}_1$

In this note, we calculate the Poincaré series of elliptic Weyl groups of type $A^{(1,1)}_1$ and $A^{(1,1) \ast}_1$ and show that they have a simple expression.