Q指数

来自testwiki

跳转到导航跳转到搜索

Template:No footnotes

animation of q-exponential

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}) \dots (1 - q)}

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

{(\frac{d}{d z})}_{q} e_{q} (z) = e_{q} (z)

{(\frac{d}{d z})}_{q} z^{n} = z^{n - 1} \frac{1 - q^{n}}{1 - q} = [n]_{q} z^{n - 1} .

关系式

当 $q < 1$

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} ϕ_{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

Template:Q超几何函数

检索自“https://zh.wiki.beta.math.wmflabs.org/w/index.php?title=Q指数&oldid=9146”

分类：

Q-模拟