$\mu$-deformation of numbers Has anyone ever encountered such a deformation of natural numbers
$$
[n]_\mu=\frac{n}{1+\mu \cdot n}?
$$
I have found a lot  identities with these $\mu$-numbers, for example:
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} k! [2]_k!=\frac{1}{2} [2]_n,
$$
$$
\sum_{k=0}^\infty \binom{2n}{k} \frac{[2]_k}{4^k}=\pi,
$$
$$
\sum_{k=0}^n \binom{1+kz}{k} \binom{(n-k)z}{n-k} \,[z]_k=z \binom{1+nz}{n}.  
$$
I hope  there is a literature on this  item.
Any reference, please.
 A: A strongly related notation can be found in Summa Summarum by M. E. Larsen, provided we consider $\mu$ as some kind of step-width. In chapter 1 Notation he introduces in section 1.2 Generalized Factorials as follows:

from section 1.2: The factorial $[x,d]_n$ is defined for any number $x\in\mathbb{C}$, any step size $d\in\mathbb{C}$ and any length $n\in\mathbb{Z}$, except for $-x\in\{d,2d,\ldots,-nd\}$ by
\begin{align*}
\color{blue}{[x,d]_n:=\begin{cases}
\prod_{j=0}^{n-1}(x-jd)&\qquad n\in\mathbb{N}\\
1&\qquad n=0,\\
\prod_{j=1}^{-n}\frac{1}{x+jd}&\qquad -n\in\mathbb{N},-x\not\in\{d,2d,\ldots,-nd\}
\end{cases}}
\end{align*}
As special cases, we remark that
\begin{align*}
[x,0]_n=x^n\qquad n\in\mathbb{Z},
\end{align*}

OPs notation $[n]_\mu=\frac{n}{1+\mu \cdot n}$ is written using Larsen's notation as
\begin{align*}
\color{blue}{[n]_\mu}=\frac{n}{1+\mu \cdot n}=\frac{1}{\frac{1}{n}+\mu}
=\prod_{j=1}^{1}\frac{1}{\frac{1}{n}+j\cdot\mu}=\color{blue}{\left[\frac{1}{n},\mu\right]_{-1}}
\end{align*}
Examples:
Elementary properties are stated for instance in section 2.1
\begin{align*}
[x,d]_k&=1/[x-kd,d]_{-k}\qquad\qquad\qquad\qquad\quad\ \  x,d\in\mathbb{C},k\in\mathbb{Z}\tag{2.3}\\
[x,d]_k&=[x-d,d]_k+kd[x-d,d]_{k-1}\qquad\qquad x,d,\in\mathbb{C},k\in\mathbb{Z}\tag{2.4}
\end{align*}
Two out of many (some hundreds) formulas are these finite versions of Whipple's formulas:
\begin{align*}
\sum_{k=0}^n&\binom{n}{k}[-n-1]_k[-a]_k[b+n-1]_{n-k}[2a-b+n]_{n-k}(-1)^k\\
&=[b+n-2,2]_n[2a-b+n-1,2]_n\tag{9.39}\\
\sum_{k=0}^n&\binom{n}{k}[a-1]_k[-a]_k[b+n-1]_{n-k}[-b-n]_{n-k}(-1)^k\\
&=[a-b-1,2]_n[-a-b,2]_n\tag{9.40}
\end{align*}
where he uses the short-hand notation $[x]_n:=[x,1]_n$ defined in section 1.2.
