Q导数

Q导数也称为杰克逊导数，乃是一般导数的Q模拟，由英国数学家 Template:Le创立。

定义

函数f(x)的q-导数定义如下：

{(\frac{d}{d x})}_{q} f (x) = \frac{f (q x) - f (x)}{q x - x} .

或书写为 $D_{q} f (x)$ .

D_{q} = \frac{1}{x} \frac{q^{\frac{d}{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：

D_{q} (f (x) + g (x)) = D_{q} f (x) + D_{q} g (x) .

D_{q} (f (x) g (x)) = g (x) D_{q} f (x) + f (q x) D_{q} g (x) = g (q x) D_{q} f (x) + f (x) D_{q} g (x) .

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

若 $g (x) = c x^{k}$ . 则

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).

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

其中 $[n]_{q}$ 是n的 q括号

并且 $\lim_{q \to 1} [n]_{q} = n$ .

一个函数的n阶导数为：

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

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}!}

$D_{q} s i n (x) = \frac{\sin (q x) - \sin (x)}{(q - 1) x}$

$D_{q} t a n h (x) = \frac{\tanh (q x) - \tanh (x)}{(q - 1) x}$

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

Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8