Q指数

Q指数是指数函数的Q模拟，定义如下

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

animation of q-exponential

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

是 Q阶乘幂

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

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

关系式

当 ${\displaystyle q<1}$

${\displaystyle e_{q}(z)=E_{q}(z(1-q)).}$

其中, ${\displaystyle E_{q}(t)}$ 是基本超几何函数的特例:

${\displaystyle E_{q}(z)=\;_{1}\phi _{0}(0;q,z)=\prod _{n=0}^{\infty }{\frac {1}{1-q^{n}z}}~.}$

参考文献

• F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538

• Gasper G., and Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, 2004, ISBN 0521833574