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) harvtxt error: no target: CITEREFChungChungNamKang1994 (help)).

Definition

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 ${\displaystyle \to {\frac {d}{dx}}}$ 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 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).

Relationship to ordinary derivatives

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]:

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

Higher order ${\displaystyle q}$-derivatives

Th 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]).

Generalizations

Post Quantum Calculus

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

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

Hahn difference

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

${\displaystyle \beta }$-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.

See also

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

References

1. F. H. Jackson (1908), On ${\displaystyle q}$-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46, 253-281.
2. Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
3. Ernst, T. (2012). A comprehensive treatment of ${\displaystyle q}$-calculus. Springer Science & Business Media.
4. Koepf, Wolfram. (2014). Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities. 10.1007/978-1-4471-6464-7.
5. Koepf, W., Rajković, P. M., & Marinković, S. D. (2007). Properties of ${\displaystyle q}$-holonomic functions.
6. Annaby, M. H., & Mansour, Z. S. (2008). ${\displaystyle q}$-Taylor and interpolation series for Jackson ${\displaystyle q}$-difference operators. Journal of Mathematical Analysis and Applications, 344(1), 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. Springer Optimization and Its Applications, vol 138. Springer.
8. Duran, U. (2016). Post Quantum Calculus, M.Sc. Thesis in Department of Mathematics, University of Gaziantep Graduate School of Natural & Applied Sciences.
9. Hahn, W. (1949). Math. Nachr. 2: 4-34.
10. Hahn, W. (1983) Monatshefte Math. 95: 19-24.
11. Foupouagnigni, M.: Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre-Freud equations for the recurrence coefficients. Ph.D. Thesis, Universit´e Nationale du B´enin, B´enin (1998).
12. Kwon, K., Lee, D., Park, S., Yoo, B.: KyungpookMath. 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, A., Sarhan, A., Shehata, E., & Aldwoah, K. (2015). A General Quantum Difference Calculus. Advances in Difference Equations, 2015(1), 182.

• Exton, H. (1983), ${\displaystyle q}$-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
• Chung, K. S., Chung, W. S., Nam, S. T., & Kang, H. J. (1994). New ${\displaystyle q}$-derivative and ${\displaystyle q}$-logarithm. International Journal of Theoretical Physics, 33, 2019-2029.

Further reading

• J. Koekoek, R. Koekoek, A note on the q-derivative operator, (1999) ArXiv math/9908140
• Thomas Ernst, The History of q-Calculus and a new method,(2001),