animation of q-exponential
Q指數是指數函數的Q模擬,定義如下
![{\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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fae3860d4b35827d5c7a220c0b7f5f5f6d93b3f)
其中
是 Q階乘冪
![{\displaystyle \left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11dad1ef314348eda0c93effeaf6a67d529b0303)
![{\displaystyle \left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5454c7904c36ab46a23fd93c086675aa4f7f618)
關係式[編輯]
當
![{\displaystyle e_{q}(z)=E_{q}(z(1-q)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c991c364f53f1b4688fbc24daa13d992328a3cf3)
其中,
是基本超幾何函數的特例:
![{\displaystyle E_{q}(z)=\;_{1}\phi _{0}(0;q,z)=\prod _{n=0}^{\infty }{\frac {1}{1-q^{n}z}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bff81a709c7bd51782fe283e9706df665ad9115)
參考文獻[編輯]
- F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.
- Gasper G., and Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, 2004, ISBN 0521833574