# Questions tagged [q-series]

Questions that are based on, use, or include the q-series in their content or solutions.

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### Need help in implementing q-SeriesToq-Product in Mathematica

In Mathematica Guidebook for symbolic computations (https://www.amazon.com/dp/0387950206/wolframresearch-20), in the Exercises, 30 (c)(p. 359), there is a question: I have no clue how to implement ...
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### Proving Gauss’s summation theorem for the $q$-binomial coefficients

I am following Warren P. Johnson's "An Introduction to q-analysis". We are supposed to prove $$\binom{n+1}{k+1}_q=\sum_{m=k}^{n}q^{m-k}\binom{m}{k}_q$$ Here is my attempt. We start off by ...
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### An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook

While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
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### Ramanujan's identity concerning a quotient of Dedekind's eta functions

In his paper On certain Arithmetical Functions (published in Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, pp. 159-184) Ramanujan presents the following identities (as if ...
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### An identity related to $q$-series

While studying Ramanujan's theta functions, I encountered a q-series $(q;q)_\infty^2\phi(q)$. I calculated the first few terms of $(q;q)_\infty^2\phi(q)$ and observed that it seems to have the ...
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### Closed form on variation of q-binomial theorem

The q-binomial theorem states that $$\sum_{n=0}^{\infty}\frac{\left(a;q\right)_n}{\left(q;q\right)_n}z^n =\frac{\left(az;q\right)_\infty}{\left(z;q\right)_\infty}$$. Is there a similar closed-form ...
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### Ramanujan Identity Proof

Ramanujan defined a now famous q-series as $$\sum_{n=-\infty}^{\infty}q^{n^2} = \left(-q;q^2\right)^2_{\infty}\left(q^2;q^2\right)_{\infty}$$ I wanted to prove this identity but I wasn't sure where to ...
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### Relation between Rogers Ramanujan continued fraction and $j$-invariant

While going through this answer I found an interesting but slightly complicated relation between Rogers-Ramanujan continued fraction and the j-invariant. I would like to know an elementary proof of ...
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### For $|q|<1$, the function $\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on $|z|<1$.

I want to prove that for $|q|<1$, the function $f(z):=\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on the set $\{z:|z|<1\}$. My approach: We consider the sequence of functions $\{f_n\}$ ...
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### Is this sum constant for n?

Hi I can prove that this sum is constant in $n\in \mathbb{N}$. However my proof is very long (a few pages with probability involved). Does anyone see a simple proof. The sum in question is (a q-series ...
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### help with an infinite series, hypergeometric function

Hi I am doing some work in interacting particle systems. I have this sum $$\sum_{k=0}^{\infty} r^{\frac{k}{2}(2m-1-k)}$$ where $m$ is some integer and $r>1$ is real. I don't how to work this ...
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