q-derivative

In mathematics, in the area of combinatorics and quantum calculus, 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. For other forms of q-derivative, see Chung et al. (1994).

Definition

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 $D_{q}f(x)$. 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, $D_{q}\to {\frac {d}{dx}}$ as $q\to 1$.

It is manifestly linear,

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

It has a 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 $g(x)=cx^{k}$. 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 e_q(x).

Relationship to ordinary derivatives

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:

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

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:

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

Higher order q-derivatives

The following representation for higher order $q$-derivatives is known:

$D_{q}^{n}f(x)={\frac {1}{(1-q)^{n}x^{n}}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}_{q}q^{{\binom {k}{2}}-(n-1)k}f(q^{k}x).$

${\binom {n}{k}}_{q}$ is the $q$-binomial coefficient. By changing the order of summation as $r=n-k$, we obtain the next formula:

$D_{q}^{n}f(x)={\frac {(-1)^{n}q^{-{\binom {n}{2}}}}{(1-q)^{n}x^{n}}}\sum _{r=0}^{n}(-1)^{r}{\binom {n}{r}}_{q}q^{\binom {r}{2}}f(q^{n-r}x).$

Higher order $q$-derivatives are used to $q$-Taylor formula and the $q$-Rodrigues' formula (the formula used to construct $q$-orthogonal polynomials).

Generalizations

Post Quantum Calculus

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:

$D_{p,q}f(x):={\frac {f(px)-f(qx)}{(p-q)x}},\quad x\neq 0.$

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):

$D_{q,\omega }f(x):={\frac {f(qx+\omega )-f(x)}{(q-1)x+\omega }},\quad 0<q<1,\quad \omega >0.$

When $\omega \to 0$ this operator reduces to $q$-derivative, and when $q\to 1$ it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.

β-derivative

$\beta$-derivative is an operator defined as follows:

$D_{\beta }f(t):={\frac {f(\beta (t))-f(t)}{\beta (t)-t}},\quad \beta \neq t,\quad \beta :I\to I.$

In the definition, $I$ is a given interval, and $\beta (t)$ is any continuous function that strictly monotonically increases (i.e. $t>s\rightarrow \beta (t)>\beta (s)$). When $\beta (t)=qt$ then this operator is $q$-derivative, and when $\beta (t)=qt+\omega$ this operator is Hahn difference.

Applications

The q-calculus has been used in machine learning for designing stochastic activation functions.

See also

• Derivative (generalizations)
• Jackson integral
• Q-exponential
• Q-difference polynomials
• Quantum calculus
• Tsallis entropy

Bibliography

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