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).

The q-derivative of a function f(x) is defined as^[1][2][3]

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

It is also often written as ${\displaystyle D_{q}f(x)}$. 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

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

which goes to the plain derivative, ${\displaystyle D_{q}\to {\frac {d}{dx}}}$ as ${\displaystyle q\to 1}$.

It is manifestly linear,

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

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

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

Similarly, it satisfies a quotient rule,

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

There is also a rule similar to the chain rule for ordinary derivatives. Let ${\displaystyle g(x)=cx^{k}}$. Then

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

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

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:^[2]

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

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

The n-th q-derivative of a function may be given as:^[3]

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

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

${\displaystyle \displaystyle D_{q}(f(x))=\sum _{k=0}^{\infty }{\frac {(q-1)^{k}}{(k+1)!}}x^{k}f^{(k+1)}(x).}$

A q-analog of the Taylor expansion of a function about zero follows:^[2]

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

The following representation for higher order ${\displaystyle q}$-derivatives is known:^[4][5]

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

${\displaystyle {\binom {n}{k}}_{q}}$ is the ${\displaystyle q}$-binomial coefficient. By changing the order of summation as ${\displaystyle r=n-k}$, we obtain the next formula:^[4][6]

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

Higher order ${\displaystyle q}$-derivatives are used to ${\displaystyle q}$-Taylor formula and the ${\displaystyle q}$-Rodrigues' formula (the formula used to construct ${\displaystyle q}$-orthogonal polynomials^[4]).

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:^[7][8]

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

Wolfgang Hahn introduced the following operator (Hahn difference):^[9][10]

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

When ${\displaystyle \omega \to 0}$ this operator reduces to ${\displaystyle q}$-derivative, and when ${\displaystyle q\to 1}$ it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.^[11][12][13]

${\displaystyle \beta }$-derivative is an operator defined as follows:^[14][15]

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

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

The q-calculus has been used in machine learning for designing stochastic activation functions.^[16]

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