# Questions tagged [q-series]

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

78 questions
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### Partial sum formula and indefinite integral of the following factorial/Lambert reciprocal function?

function is as follows: $$\frac 1{(2m)!(1-q^{2m})}$$ $$s.t. (m, q) \in \mathbb N$$ I ask this because of some difficulties encountered in a previous question I'd asked, linked here: Previous ...
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### Partial sum formula of following series

$$\sum_{m \ge 1} \frac{(xy)^m}{(2m)!(1-y^m)}, \quad\text{where }x,y \in \mathbb N$$ I have, to start, J.Jacquelin's answer.
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### What is a closed form partial sum formula for the q-digamma function?

It is known that the partial sum of the digamma function can be expressed in closed form, but what of the q-digamma function? $$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$ $$q\in\mathbb N$$
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### Double sum of Lambert series: Partial sum in closed form desired!

We desire the things stated in the title for: $\sum_{k=2}^m \sum_{n=1}^{k-1} {q^n\over {1-q^n}}$ Some things I've looked into that may be of some help: The first sum is just a truncated (partially ...
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### How to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$?

I need to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$ but wolframalpha's brain is too narrow to contain the result, and I don't know any software and programming to find ...
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### Identity between $q$-series

Fix some sequence $(a_k)$ such that $a_k=R_k+q^k$ and $R_k=\frac{1}{R(q^k,q)}$,where $R(x,q)$ is a Generalized Rogers-Ramanujan continued fraction for complex number $x$ and $q=\exp{(2\pi i\tau)}$ is ...
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### q-exponential functions

We know that $$e_q(z)=\sum_{j\geq 0} \frac{z^j}{(q;q)_j}=\frac{1}{(z;q)_{\infty}}$$ where $(a;q)_{\infty}=\prod_{i=0}^{\infty}(1−aq^i)$ denotes the q-shifted factorial. The limit between the $q$-...
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### q-shifted factorial

Is their any formula to simplify $$(a,ab;q)_n$$ where $(a;q)_n=\prod_{i=0}^{n-1}(1−aq^i)$ denotes the q-shifted factorial. P.S: $a$ and $b$ are complex numbers. I'm glade for your help.
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### Does the Rogers-Ramanujan continued fraction $R(q)$ satisfy this conjectured infinite series

Given the Rogers-Ramanujan continued fraction $R(q)= \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}}$ where $q=\exp(2\pi i \tau)$, $|q|\lt1$ for the sake of brevity, let us ...
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### Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
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### a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
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### Some analogs of the pentagonal number theorem

There are the following analogs of the famous identity $$\prod_{n\geqslant1}(1-q^n)=\sum_{n\in\mathbb Z}(-1)^nq^{\frac{3n^2-n}2}.$$ Let $v_2(n)$ denote the 2-adic valuation of $n$, that is, the ...
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