# Questions tagged [q-series]

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Consider the analytic space $\mathbb{C}^{*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}... 0answers 59 views ### Concerning the coefficient [q^n]\sum_{n\ge1}\frac{(aq)^n}{(1-bq^n)^2} I posted this on MSE, and not even @ParamanandSingh could answer, so I thought I should post it here Background: While trying to answer this question, I came up with a question of my own. Let |a|,|b|,... 1answer 438 views ### q-series identity related to Jackson-Slater, proof required The question: I have been trying to prove the following q-series identity for quite some time now:$$ \sum_{k \geq 0} \frac{q^{2k^2}}{(q)_{2k}} = \sum_{m,k \geq 0} \frac{q^{m^2 + 3k m + 4k^2}}{(q)...
In this paper, Shanks uses the following formula: $$\sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a ...