Keywords

• quantum calculus
• Jackson derivative
• q-monogenic function

Abstract $q$-calculus or quantum calculus is a calculus based on a difference quotient instead of the limit of a difference quotient, i.e. the $q$-derivative $D_q$ is defined as:

$D_q f (x) = \dfrac{f (qx) − f (x)}{qx − x},\quad −1 < q < 1$.

This derivative is also called Jackson derivative and the one-dimensional Jackson calculus [2], [3] is based on that derivative. $q$-calculus has a wide ranged number of applications, for example $q$-polynomials, $q$-special functions, $q$-difference equations and in physics [1]. Because $D_q (xn) = [n]_q x^{n−1}$, where $[n]_q = \frac{1−q^n}{q−1}$, we call $[n]_q$ $q$-numbers. We consider the $q$-Dirac operator

$D_q = \displaystyle\sum_{j=1}^m e_j \partial^q_j$, where $\partial^q_j f(x) = \dfrac{f (x_1, \ldots, x_{j−1}, qx_j , x_{j+1}, \ldots , x_m) − f (x)}{qx_j − x_j}$.

We construct some $q$-monogenic functions, consider the $q$-Euler and $q$-Gamma operator, Fischer decomposition and Cauchy-Kovalevskaja extension theorem. Our approach is different from the one in [4].

References

[1] T. Ernst. A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel, 2012. doi:10.1007/978-3-0348-0431-8.

[2] F. H. Jackson. On q-Functions and a certain Difference Operator. In: Transactions of the Royal Society of Edinburgh 46.2 (1909), 253–281. doi:10.1017/S0080456800002751.

[3] V. Kac and P. Cheung. Quantum Calculus. Universitytext. Springer, 2002.

[4] K. Coulembier and F. Sommen. Operator Identities in q-Deformed Clifford Analysis. In: Advances in Applied Clifford Algebras 21.4 (2011), pp. 677–696. doi:10.1007/s00006-011-0281-9