Q导数

函数f(x)的q-导数定义如下：

${\displaystyle \left({\frac {d}{dx}}\right)_{q}f(x)={\frac {f(qx)-f(x)}{qx-x}}.}$ 

或书写为 ${\displaystyle D_{q}f(x)}$ .

${\displaystyle D_{q}={\frac {1}{x}}~{\frac {q^{d~~~ \over d(\ln x)}-1}{q-1}}~,}$ 

当as q → 1时，化为寻常的导数, → ^d⁄_dx,

q-导数算符是一个线性算子：

${\displaystyle \displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.}$ 

${\displaystyle \displaystyle D_{q}(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)D_{q}g(x).}$ 

${\displaystyle \displaystyle D_{q}(f(x)/g(x))={\frac {g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)}},\quad g(x)g(qx)\neq 0.}$ 

若 ${\displaystyle g(x)=cx^{k}}$ . 则

${\displaystyle \displaystyle D_{q}f(g(x))=D_{q^{k}}(f)(g(x))D_{q}(g)(x).}$ 

q-导数 的本征值是q-指数 e_q(x).

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

其中 ${\displaystyle [n]_{q}}$  是n的 q括号

并且 ${\displaystyle \lim _{q\to 1}[n]_{q}=n}$  .

一个函数的n阶导数为：

${\displaystyle (D_{q}^{n}f)(0)={\frac {f^{(n)}(0)}{n!}}{\frac {(q;q)_{n}}{(1-q)^{n}}}={\frac {f^{(n)}(0)}{n!}}[n]_{q}!}$ 

${\displaystyle f(z)=\sum _{n=0}^{\infty }f^{(n)}(0)\,{\frac {z^{n}}{n!}}=\sum _{n=0}^{\infty }(D_{q}^{n}f)(0)\,{\frac {z^{n}}{[n]_{q}!}}}$ 

${\displaystyle D_{q}sin(x)={\frac {\sin \left(qx\right)-\sin \left(x\right)}{\left(q-1\right)x}}}$ 

${\displaystyle D_{q}tanh(x)={\frac {\tanh \left(qx\right)-\tanh \left(x\right)}{\left(q-1\right)x}}}$ 

• Derivative (generalizations)（英语：Derivative (generalizations)）
• 杰克逊积分（英语：Jackson integral）
• Q指数
• Q-差值积分（英语：Q-difference polynomial）
• 量子微积分（英语：Quantum calculus）
• Tsallis entropy（英语：Tsallis entropy）

• 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, 编辑
  • J. Koekoek, R. Koekoek, A note on the q-derivative operator（页面存档备份，存于互联网档案馆）, (1999) ArXiv math/9908140
  • Thomas Ernst, The History of q-Calculus and a new method,(2001),