Quantum integer


context $f:{\mathbb N}\to{\mathbb R}$
definiendum $[n]_q \in \mathrm{it}$
inclusion $[n]_q:{\mathbb N}\to{\mathbb C}^*\to{\mathbb R}$
definition $[n]_q:=q^{-f(n)/2}\frac{1-q^n}{1-q}$


These are $q$-deformations of integers, so that arithmetic coincides at $q=1$.

$[n]_{q} = q^{-f(n)/2}q^{-1}\sum_{k=1}^n q^k = n+\tfrac{n}{2}(n-1-f(n))\cdot(q-1)+\mathcal{O}\left((q-1)^2\right)$

In fact this doesn't require $n$ to be an integer.

The case $f=0$ is often considered.

Quantum aspect: $f=n-1$ gives

$[n]_{q^2} = n + \mathcal{O}\left((q-1)^2\right)$.

(The $q^2$ isn't necessary.) In the imaginary direction, $q\propto\mathrm{e}^{i\varphi}$, this corresponds to $\lim_{\varphi\to 0}\frac{\sin(n\varphi)}{\sin(\phi)}=n$. With $q=r\mathrm{e}^{i\varphi}$, along the positive real axis number $[n]_q$ is a valley with bottom at $q=1$, where $[n]_{1}=n$, and along $\varphi$ you have harmonic oscillations with period depending on $n$.

I might change the exponent in $-f(n)/2$ to something else later
I see one can also capture it as
K[q_, a_, b_, c_, d_] = q^(b - c) (1 - q^(a + b))/(1 - q^(1 + c + d))
and then
K[q, n, 0, 0, 0]
K[q, -3 n, n, 1, -4]


Wikipedia: q-analog



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