查看“︁Q指数”︁的源代码

{{no footnotes|time=2015-04-06T01:31:32+00:00}}
[[File:Qexp.gif|thumb|300px|animation  of  q-exponential]]
'''Q指数'''是指数函数的[[Q-模拟|Q模拟]]，定义如下

:<math>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)}</math>

其中<math>(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)</math>

是 [[Q阶乘幂]]

:<math>\left(\frac{d}{dz}\right)_q e_q(z) = e_q(z)</math>

:<math>\left(\frac{d}{dz}\right)_q z^n = z^{n-1} \frac{1-q^n}{1-q}
=[n]_q z^{n-1}.</math>

==关系式==
当 <math>q<1</math>

:<math>e_q(z) = E_q(z(1-q)).</math>

其中, <math>E_q(t)</math> 是[[基本超几何函数]]的特例: 

:<math>E_q(z) = \;_{1}\phi_0 (0;q,z) = \prod_{n=0}^\infty 
\frac {1}{1-q^n z} ~. </math>

==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


[[Category:Q-模拟]]
{{q超几何函数}}