I have to show the Jacobi Quintuple Product Identity $$\prod_{n = 1}^{\infty} (1-q^n)(1- \zeta q^{n-1})(1-\zeta^{-1} q^{n})(1-\zeta^{2} q^{2n-1})(1-\zeta^{-2}q^{2n-1}) = \sum_{n \in \mathbb{Z}} q^{\frac{3}{2}n^2-\frac{1}{2}n} \zeta^{3n} (1-\zeta q^n)$$
$q = e^{2 \pi i \tau}$ $ \zeta = e^{2 \pi iz}$
For this i have: $$f(z, \tau)= \nu_{0,1/2} (z, \tau ) \nu_{1/2,1/2} (2z, 2 \tau )$$
i muss show, that $$f(z+1, \tau) =f(z ,\tau) , \,\,\, f(z+\tau, \tau) =e^{-3 \pi i \tau -6 \pi iz} f(z,\tau)$$
$$ => f( z, \tau) = \sum_{k=0}^{2}c_k(\tau) \nu_{k/3,0} (3z, 3\tau)$$ whit $c_k (\tau)= e^{-\pi\tau k^2/3} \int_{0 }^{1} f(z, \tau) e^{-2 \pi ikz} dz $
Than i muss show, that $c_0=0$ and $c_2 = -c_1$
=>than i muss show, that $c_1(\tau) = e^{\pi i/3} \nu_{1/6,1/2} (0, 6 \tau)$
=>now i have a identity with theta series
Can you help me?
Thanks in advance