q-derivative

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In mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration.

Definition[edit]

The q-derivative of a function f(x) is defined as

\left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}.

It is also often written as D_qf(x). The q-derivative is also known as the Jackson derivative.

Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator

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

which goes to the plain derivative, → ddx, as q → 1.

It is manifestly linear,

\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~.

It has product rule analogous to the ordinary derivative product rule, with two equivalent forms

\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).

Similarly, it satisfies a quotient rule,

\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.

There is also a rule similar to the chain rule for ordinary derivatives. Let g(x) = c x^k. Then

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

The eigenfunction of the q-derivative is the q-exponential eq(x).

Relationship to ordinary derivatives[edit]

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:

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

where [n]_q is the q-bracket of n. Note that \lim_{q\to 1}[n]_q = n so the ordinary derivative is regained in this limit.

The n-th q-derivative of a function may be given as:

(D^n_q f)(0)=
\frac{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}= 
\frac{f^{(n)}(0)}{n!} [n]_q!

provided that the ordinary n-th derivative of f exists at x = 0. Here, (q;q)_n is the q-Pochhammer symbol, and [n]_q! is the q-factorial. If f(x) is analytic we can apply the Taylor formula to the definition of D_q(f(x)) to get

\displaystyle D_q(f(x)) = \sum_{k=0}^{\infty}\frac{(q-1)^k}{(k+1)!} x^k f^{(k+1)}(x).

A q-analog of the Taylor expansion of a function about zero follows:

f(z)=\sum_{n=0}^\infty f^{(n)}(0)\,\frac{z^n}{n!} = \sum_{n=0}^\infty (D^n_q f)(0)\,\frac{z^n}{[n]_q!}

See also[edit]

References[edit]

  • F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.
  • Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8

Further reading[edit]