Hyperbolic function theory studies Clifford algebra-valued functions defined in the hyperbolic half-space. A central concept in this theory is that of \(\alpha \) -hypermonogenic functions, which, for the real parameter value \(\alpha = 0\) , reduce to the classical monogenic functions. In this paper, we extend integral formulas to general Clifford algebra-valued functions by deriving the so-called Borel–Pompeiu formulas. The corresponding kernels are computed explicitly, providing a detailed analysis of the integral representations. Additionally, we extend the theory to the lower half-space and address the problem posed by the singular hypersurface \(x_n = 0\) . We investigate how functions defined in the upper half-space can be transformed into reflected functions in the lower half-space while preserving hypermonogenicity. Furthermore, we consider a similar problem for \(\alpha \) -hyperbolically harmonic functions. Our work builds on previous research and contributes to the development of integral representations and function theory in hyperbolic geometry.
