Questions tagged [q-series]

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Analytic function with q- difference equation involving theta

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}...
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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|,...
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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)...
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Finite version special case Jacobi triple product formula

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 ...