# Questions tagged [q-series]

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### Reference for modularity of the Andrews–Gordon–Rogers–Ramanujan identities?

The right-hand side of the identity https://mathworld.wolfram.com/Andrews-GordonIdentity.html is a $q$-series $\frac{(q^i,q^{2k+1-i},q^{2k+1};q^k)_\infty}{(q;q)_\infty}$; is there a reference of its ...
Recall that a $q$-Pochhammer symbol is defined as $$(x)_n = (x;q)_n := \prod_{l=0}^{n-1}(1-q^l x).$$ I found the following curious $q$-series identity that seems to hold for any $n\geq 0$: $$(-1)^{... 6 votes 0 answers 93 views ### q-binomial-like series with exponentials defining probability distribution Recently I encountered the series$$f(d) = \frac{1}{(t;t)_\infty} \sum_{k=0}^\infty \frac{(-1)^k t^{\binom{k}{2}}}{(t;t)_k} e^{-t^{d-k}}$$where (t;t)_n = \prod_{i=1}^n (1-t^i), and 0 < t < 1... 0 votes 0 answers 114 views ### Addition formulas for q-analogs of trigonometric functions Sine and Cosine functions possess notable formulas for addition of angles$$ \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b). $$One can ... 1 vote 1 answer 116 views ### Analytic function with q- difference equation involving theta Consider the analytic space \mathbb{C}^{*} with coordinate z. Let q be some parameter with |q|<1 and define the analytic function$$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}... 