In this paper, we present explicit representations for the Neumann boundary value problem associated with the Bitsadze equation on the upper half-plane. Additionally, we provide solutions to the inhomogeneous polyanalytic equation that arises from Neumann and (n–1) Dirichlet boundary conditions on the upper half-plane H. The analysis employs Cauchy–Pompeiu representations and extensions of Gauss’s theorem to derive precise formulations under the specified boundary conditions.