q-Dirac operator; quantum Clifford analysis; Clifford algebra

In this paper, we provide the foundations of quantum Clifford analysis in q-commutative variables with symmetric difference operators. We consider a q-Dirac operator on the quantum Euclidean space that factorizes the U_q(o)-invariant Laplacian ∆_q. Due to the non-commutativity of the multiplication, we need a special Clifford algebra. We define q-monogenic functions as null solutions of the q-Dirac operator and q-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree k.