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[1][2][3]

\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, 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) = 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

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

\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:[3]

(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:[2]

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}.

Higher order q-derivatives

The following representation for higher order q-derivatives is known:[4][5]

D_q^nf(x)=\frac{1}{(1-q)^nx^n}\sum_{k=0}^n(-1)^k\binom{n}{k}_q q^{\binom{k}{2}-(n-1)k}f(q^kx).

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

D_q^nf(x)=\frac{(-1)^n q^{-\binom{n}{2}}}{(1-q)^nx^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[4]).

Generalizations

Post Quantum Calculus

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

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):[9][10]

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

When \omega\to0 this operator reduces to q-derivative, and when q\to1 it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.[11][12][13]

β-derivative

\beta-derivative is an operator defined as follows:[14][15]

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.[16]

See also

Citations

  1. ^ Jackson 1908, pp. 253–281.
  2. ^ Kac & Pokman Cheung 2002.
  3. ^ Ernst 2012.
  4. ^ Koepf 2014.
  5. ^ Koepf, Rajković & Marinković 2007, pp. 621–638.
  6. ^ Annaby & Mansour 2008, pp. 472–483.
  7. ^ Gupta V., Rassias T.M., Agrawal P.N., Acu A.M. (2018) Basics of Post-Quantum Calculus. In: Recent Advances in Constructive Approximation Theory. SpringerOptimization and Its Applications, vol 138. Springer.
  8. ^ Duran 2016.
  9. ^ Hahn, W. (1949). Math. Nachr. 2: 4-34.
  10. ^ Hahn, W. (1983) Monatshefte Math. 95: 19-24.
  11. ^ Foupouagnigni 1998.
  12. ^ Kwon, K.; Lee, D.; Park, S.; Yoo, B.: Kyungpook Math. J. 38, 259-281 (1998).
  13. ^ Alvarez-Nodarse, R.: J. Comput. Appl. Math. 196, 320-337 (2006).
  14. ^ Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.
  15. ^ Hamza et al. 2015, p. 182.
  16. ^ Nielsen & Sun 2021, pp. 2782–2789.

Bibliography