Q指数

Q指数是指数函数的Q模拟，定义如下

: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(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)

是 Q阶乘幂

:\left(\frac{d}{dz}\right)_q e_q(z) = e_q(z)

:\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 q

:e_q(z) = E_q(z(1-q)).

其中, E_q(t) 是基本超几何函数的特例:

: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