# Questions tagged [q-analogs]

Use this tag for questions pretaining to q-analogs of functions, for example q-Binomials, $q$-derivatives, the q-theta function, the q-Pochhammer symbol, etc.

75 questions
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### A closed form of $q$-analog finite sum

Let $[n]:=\frac{1-q^n}{1-q}$. I wish to find a closed form of the following q-sum $$I(p):=\sum\limits_{n_1+\cdots+n_p=n\atop n_1,\ldots,n_p\ge 1}\ \ \ \frac{1}{[n_1]\cdots [n_p]}=？$$ For example, we ...
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### $q$-analogs confusion in some steps

I am understanding the proof of the general version of the Jacobi triple product, that is $$\prod_{k=1}^{\infty}(1+xq^{2k-2})=\sum_{k=0}^{\infty}\frac{q^{k(k-1)}}{(q^2)_k}x^k$$ In the proof to this ...
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### Importance of $q$-analog

I am currently studying q-analog, but I was actually confused on what its actual purpose is. Like I see all these manipulations using $q$, but I have little idea on what they represent. Sure the ...
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### Stanley's Enumerative Combinatorics Problem 125 Page 140

The problem states: Find the number $f(n)$ of binary sequences $w = a_1a_2...a_k$ (where k is arbitrary) such that $a_1 = 1$, $a_k = 0$, and $inv(w) = n$. For instance, $f(4) = 5$, corresponding to ...
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### Prove identity involving the Tsallis q-logarithm

The natural logarithm and the exponential can both be generalized to a called q-logarithms and q-exponentials.those functions are defined as follows: \begin{eqnarray} \log_q(x) &:=& \frac{x^{1-...
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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: \begin{equation} \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\...