Conjugate (1/q, q)-Harmonic Polynomials in q-Clifford Analysis

We consider the problem of constructing a conjugate (1/q, q)-harmonic homogeneous polynomial Vk of degree k to a given (1/q, q)-harmonic homogeneous polynomial Uk of degree k. The conjugated harmonic polynomials Vk and Uk are associated to the (1/q, q)-monogenic polynomial F=Uk+e¯0Vk. We investigate conjugate (1/q, q)-harmonic homogeneous polynomials in the setting of q-Clifford analysis. Starting from a given (1/q, q)-harmonic polynomial Uk of degree k, we construct its conjugate counterpart Vk, such that the Clifford-valued polynomial F=Uk+e0Vk is (1/q, q)-monogenic, i.e., a null solution of a generalized q-Dirac operator. Our construction relies on a combination of Jackson-type integration, Fischer decomposition, and the resolution of a q-Poisson equation. We further establish existence and uniqueness results, and provide explicit representations for conjugate pairs, particularly when Uk is real-valued.