The Poincaré series of the central extension W˜ of the elliptic Weyl group of type A1(1,1)

We calculate the Poincaré series (also called growth series) of the central extension of the elliptic Weyl group of type A1(1,1) with generators wi, wi∗(i=0,1), which is a central extension of the Weyl group associated to the elliptic root system of type A1(1,1).