Cfrgtkky

In this post, the Kac character formula for affine $mathfrak{su}(2)_k$
$$chi_{ell}^{(k)}(tau,z) = frac{Theta_{ell+1,k+2}(tau,z)-Theta_{-ell-1,k+2}(tau,z)}{Theta_{1,2}(tau,z)-Theta_{-1,2}(tau,z)}$$
with
$$Theta_{ell,k}(tau,z)=sum_{nin mathbb{Z}+frac{ell}{2k}} q^{kn^2}y^{kn}, ,qquad (q=e^{2pi itau},,:y=e^{2pi iz})$$
is put in a nice form:
$$chi_{ell}^{(k)}(tau,z)=q^{m_{ell}}frac{sum_{ninmathbb{Z}}frac{sinleft[(ell+1+2n(k+2))pi zright]}{sin(pi z)}q^{n(ell+1)+n^2(k+2)}}{prod_{n>0}(1-q^n)(1-q^n y)(1-q^n y^{-1})}, .$$

I'm running into some trouble getting this expression. I'm guessing the Jacobi triple product identity is used for the denominator, but when I try to compute it I get
begin{align*} Theta_{1,2}(tau,z)-Theta_{-1,2}(tau,z)&=&sum_{ngeqslant 0}q^{2(n+1/4)^2}y^{2(n+1/4)}-sum_{ngeqslant 0}q^{2(n-1/4)^2}y^{2(n-1/4)}\&=& prod_{n>0}(1+q^n y^2)(1+q^n y^{-2})(1-q^{n+2}),.end{align*}
I'm running into the same kind of trouble for the numerator, so I suspect I'm not using the $Theta$ functions right... Some more details on this derivation would be much appreciated, or references where things are done a bit more explicitly!