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.
