# Tagged Questions

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### Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
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### show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd

I would like to show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd, or preferably even more generally that $\sum_{j=0}^n (-1)^j {n \brack j }_q =\frac{1}{2}((-1)^n+1)(q;q)_{\frac{n}{2}}$. Using ...
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### About $\prod{\left(1-q^n\right)^{5}}$

Is there a result about the non-vanishing of coefficients of $$\prod_{n=1}^{+\infty}{\left(1-q^n\right)^{5}}=1-5q+5q^2+10q^3-15q^4-6q^5-5q^6+25q^7+15q^8-20q^9+\cdots \text{ ?}$$ Thanks !
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### A question on a sum of $q$-binomial coefficients

I am trying to enumerate a certain quantity and at some point I get the following sum: \sum_{i=0}^{m}{m \brack i}_q \sum_{j=0}^{n-m} q^{j(m-i)}{n-m \brack j}_q \sum_{k=0}^{r} {r\...
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### $q$-series identity

I have to prove the following identity: $$\sum_{n\geq 0} (-1)^n(2n+1)q^{\frac{n(n+1)}{2}} = (q;q)_\infty^3$$ where $(a;q)_\infty = \prod_{i\geq 0}1-aq^i$ is the $q$-Pochhammer symbol. In my notes the ...
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### Combinatorial interpretation of this identity of Gauss?

Gauss came up with some bizarre identities, namely $$\sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}.$$ How can this be interpreted combinatorially? It strikes me as being ...
### factoring infinite products of $q$-series with constant term equal to 1
I was thinking about the following infinite product: $$\prod_{n=0}^{\infty} \frac{ae^{-2n}+be^{-n}+c}{c}$$ The right way of generalizing it is to think in terms of $q$-Pochhammer symbols. If $r_{1}$ ...
### Closed form for $\sum_{m \geq 1} (-1)^m q^{m(m+1)/2 + m \Delta}$?
Is there a useful closed form for the following series ($|\Delta|$ is a small integer)? $$f(q,\Delta) =\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 + m \Delta}$$ It is a large-$n$ approximation of the ...