Hermitian and non-Hermitian Bergman kernels in octonionic analysis over concentric and sectorial ball domains

In this paper, we study the reproducing Bergman kernel on smooth bounded domains in the eight-dimensional non-associative octonionic algebra. We establish necessary and sufficient conditions on associator terms involving an intrinsic octonionic weight factor that ensure the existence of a Bergman kernel for square-integrable solutions to the octonionic Cauchy-Riemann equation. The kernel is constructed explicitly through a correspondence with the Green’s function of the domain, yielding closed-form representation formulas for concentric annuli, where the well-known formula for the solid ball is obtained as a limit case. In these cases, the kernel is Hermitian and induces the Bergman projection from the L₂ space of monogenic functions onto the Bergman space. In contrast, for half-balls, orthogonal sectors, and annular sectors, the associator condition fails, and no Hermitian Bergman kernel exists when using the standard weight factor x /| x | for the spherical setting. In such cases, we construct a unique non-Hermitian generalized kernel that reproduces all octonionic monogenic functions, though lacking Hermitian symmetry and therefore not serving as a complete L₂ -projection.