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(Redirected from Jackson derivative) Q-analog of the ordinary 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

( d d x ) q f ( x ) = f ( q x ) − f ( x ) q x − x . {\displaystyle \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 ) {\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

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

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

It is manifestly linear,

D q ( f ( x ) + g ( x ) ) = D q f ( x ) + D q g ( x )   . {\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

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 ) . {\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,

D q ( f ( x ) / g ( x ) ) = g ( x ) D q f ( x ) − f ( x ) D q g ( x ) g ( q x ) g ( x ) , g ( x ) g ( q x ) ≠ 0. {\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 g ( x ) = c x k {\displaystyle g(x)=cx^{k}} . Then

D q f ( g ( x ) ) = D q k ( f ) ( g ( x ) ) D q ( g ) ( x ) . {\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 eq(x).

Relationship to ordinary derivatives

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

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

where [ n ] q {\displaystyle _{q}} is the q-bracket of n. Note that lim q → 1 [ n ] q = n {\displaystyle \lim _{q\to 1}_{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 ) = f ( n ) ( 0 ) n ! ( q ; q ) n ( 1 − q ) n = f ( n ) ( 0 ) n ! [ n ] ! q {\displaystyle (D_{q}^{n}f)(0)={\frac {f^{(n)}(0)}{n!}}{\frac {(q;q)_{n}}{(1-q)^{n}}}={\frac {f^{(n)}(0)}{n!}}!_{q}}

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

D q ( f ( x ) ) = ∑ k = 0 ∞ ( q − 1 ) k ( k + 1 ) ! x k f ( k + 1 ) ( x ) . {\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:

f ( z ) = ∑ n = 0 ∞ f ( n ) ( 0 ) z n n ! = ∑ n = 0 ∞ ( D q n f ) ( 0 ) z n [ n ] ! q . {\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}}{!_{q}}}.}

Higher order q-derivatives

The following representation for higher order q {\displaystyle q} -derivatives is known:

D q n f ( x ) = 1 ( 1 − q ) n x n ∑ k = 0 n ( − 1 ) k ( n k ) q q ( k 2 ) − ( n − 1 ) k f ( q k x ) . {\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).}

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

D q n f ( x ) = ( − 1 ) n q − ( n 2 ) ( 1 − q ) n x n ∑ r = 0 n ( − 1 ) r ( n r ) q q ( r 2 ) f ( q n − r x ) . {\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 q {\displaystyle q} -derivatives are used to q {\displaystyle q} -Taylor formula and the q {\displaystyle q} -Rodrigues' formula (the formula used to construct q {\displaystyle 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 ) := f ( p x ) − f ( q x ) ( p − q ) x , x ≠ 0. {\displaystyle 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 , ω f ( x ) := f ( q x + ω ) − f ( x ) ( q − 1 ) x + ω , 0 < q < 1 , ω > 0. {\displaystyle D_{q,\omega }f(x):={\frac {f(qx+\omega )-f(x)}{(q-1)x+\omega }},\quad 0<q<1,\quad \omega >0.}

When ω → 0 {\displaystyle \omega \to 0} this operator reduces to q {\displaystyle q} -derivative, and when q → 1 {\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.

β-derivative

β {\displaystyle \beta } -derivative is an operator defined as follows:

D β f ( t ) := f ( β ( t ) ) − f ( t ) β ( t ) − t , β ≠ t , β : I → I . {\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, I {\displaystyle I} is a given interval, and β ( t ) {\displaystyle \beta (t)} is any continuous function that strictly monotonically increases (i.e. t > s → β ( t ) > β ( s ) {\displaystyle t>s\rightarrow \beta (t)>\beta (s)} ). When β ( t ) = q t {\displaystyle \beta (t)=qt} then this operator is q {\displaystyle q} -derivative, and when β ( t ) = q t + ω {\displaystyle \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

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

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