Q指数

Q指数是指数函数的Q模拟，定义如下

animation of q-exponential

${\displaystyle e_{q}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}(1-q)^{n}}{(q;q)_{n}}}=\sum _{n=0}^{\infty }z^{n}{\frac {(1-q)^{n}}{(1-q^{n})(1-q^{n-1})\cdots (1-q)}}}$

其中${\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)}$

是 Q阶乘幂

${\displaystyle \left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)}$

${\displaystyle \left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}.}$