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I've run across an interesting hypergeometric $q$-series that I feel must have been found before:

$\sum_{n=0}^{\infty}(-1)^n$$\frac{e^{n b y}}{\prod_{k=1}^{n}(e^{\pi k b^2}-e^{\pi k b^{-2}})} = \sum_{n=0}^{\infty}(-1)^n$$\frac{z^n}{(q^2;q^2)_n} q^{\binom{n}{2}} = \sum_{n=0}^{\infty}(-1)^n$$\frac{z^n}{(q;q)_n(-q;q)_n} q^{\binom{n}{2}} $

I've been playing around with it for a while, using the basic identities (and limits thereof) found in Gasper and Rahman but I've been unable to find a closed form for the series. I'm not sure if this is quite a research level question but I was curious if anyone had any insight into a closed form for this expression.

Thank you!

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Comparing with the modern (Gasper-Rahman) definition of the basic hypergeometric function, we have the expression

$$\sum_{n=0}^\infty\frac{(-1)^n z^n q^\binom{n}{2}}{(-q;q)_n(q;q)_n}={}_1\phi_1\left({{0}\atop{-q}}\middle|q,z\right)$$

(I assumed the omission of the indices in the $q$-Pochhammer symbols was unintentional.)

I do not know if there is a simpler closed form.

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