  This paper has a twofold purpose: on one hand we deepen the study of slice
regular functions by studying their behavior with respect to the so-called
C-property and anti-C-property. We show that, for any fixed basis of the
algebra of quaternions $\mathbb{H}$ any slice regular function decomposes into
the sum of four slice regular components each of them satisfying the
C-property. Then, we will use these results to show a reproducing property of
the Bergman kernels of the second kind.

Complex Variables