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

Abstract

We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and $U_k$ are associated to the $(1/q, q)$-mono\-genic polynomial $F = U_k + \overline{e}_0V. $ 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 $U_k$ of degree $k$, we construct its conjugate counterpart $V_k$, such that the Clifford-valued polynomial $F = U_k + e_0 V_k$ 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 $U_k$ is real-valued.