General Aspects of Jackson Calculus in Clifford Analysis

Abstract

We consider an extension of Jackson calculus into higher dimensions and specifically into Clifford analysis for the case of commuting variables. In this case, Dirac is the operator of the first q-partial derivatives (or q-differences) \({_{q}}\mathbf {\mathcal {D}}= \sum _{i=1}^n e_i\,{_{q}}\partial _i\), where \({_{q}}\partial _i\) denotes the q-partial derivative with respect to \(x_i\). This Dirac operator factorizes the q-deformed Laplace operator. Similar to the case of classical Clifford analysis, we then consider the q-deformed Euler and Gamma operators and their relations to each other. Nullsolutions of this q-Dirac equation are called q-monogenic. Using the Fischer decomposition, we can decompose the space of homogeneous polynomials into spaces of q-monogenic polynomials. Using the q-deformed Cauchy–Kovalevskaya extension theorem, we can construct q-monogenic functions. Overall, we show the analogies and the differences between classical Clifford and Jackson-Clifford analysis. In particular, q-monogenic functions need not be monogenic and vice versa.