Discrete q-Hermitian Clifford analysis

Abstract

We develop a discrete $q$-Hermitian Clifford calculus based on coordinatewise Jackson differences on multiplicative $q$-lattices. The resulting Hermitian Jackson--Dirac operators are nilpotent, factor a $q$-Laplacian, and have occupancy-dependent Euler anticommutators. These give explicit Fischer projectors and homotopies, but the local calculus is not controlled by total degree. We prove a two-face Cauchy--Kovalevskaya theorem for polynomials and extend it to a $q$-analytic Jackson--Fischer class. A divided-power conjugation recovers scalar Euler relations and transfers the classical joint Fischer decomposition and dimension formulas. The Jackson--Fischer completion is a vector-valued $q$-Fock space whose polarized nullspaces have projected reproducing kernels; the joint kernel is obtained by alternating projections. We also determine the linear symmetry group of the multiplicative lattice.