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

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10
votes
0answers
226 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
7
votes
0answers
174 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
4
votes
0answers
138 views

A general Combinatorics problem (Coefficients of the q factorial)

I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...
4
votes
0answers
63 views

Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?

In this question, user begins shows that, for each $k\in \mathbb{N}$, there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, ...
2
votes
0answers
23 views

Closed formula involving $q$-binomials

I was working on a combinatorial problem over finite fields, and the following quantity came up $$ \sum_{r=0}^k r\binom{n-k}{r}_q\binom{k}{k-r}_qq^{r^2},$$ where $k,n$ are integers such that ...
2
votes
0answers
44 views

Barnes' double gamma function versus q-gamma function

According to wikipedia, the q-analog of the gamma function is closely related to a multiple gamma function defined by Barnes. Besides the fact that they are both generalizations of the Gamma function, ...
1
vote
0answers
31 views

An identity for $q-$Fibonacci numbers if $q$ is a root of unity.

In his proof of the Rogers-Ramanujan identities I. Schur introduced two $q-$analogues of the Fibonacci numbers ${F_n}({q})$ and ${G_n}({q})$, which satisfy ${F_0}({q})=0$, ${F_1}({q})=1$ and ...
1
vote
0answers
30 views

How to prove that $\sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}=\frac{(ab;q)_n}{(q;q)_n}$?

By Cauchy identity, $${}_1\phi_0(a;—;q,z)=\sum_{n\geq0}\frac{(a;q)_n}{(q;q)_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_\infty},\quad|z|<1,|q|<1,$$ we can obtain the $q-$analogue of ...
0
votes
0answers
50 views

when cdf=( i-0.5)/n and you have a negative

I am stuck with when you set your cdf to equal $\frac{i-0.5}{n}$ for when you are plotting QQ plots. I have: $$-e^{\frac{-x^2}{2\sigma^2}}= \frac{i-0.5}{n}$$ Then I got stuck because I need to take ...