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The aim of this paper is to obtain a generalized CK-extension theorem insuperspace for the bi-axial Dirac operator. In the classical commuting case,this result can be written as a power series of Bessel type of certaindifferential operators acting on a single initial function. In the superspacesetting, novel structures appear in the cases of negative even superdimensions.In these cases, the CK-extension depends on two initial functions on which twopower series of differential operators act. These series are not only of Besseltype but they give rise to an additional structure in terms of Appellpolynomials. This pattern also is present in the structure of the Pizzettiformula, which describes integration over the supersphere in terms ofdifferential operators. We make this relation explicit by studying thedecomposition of the generalized CK-extension into plane waves integrated overthe supersphere. Moreover, these results are applied to obtain a decompositionof the Cauchy kernel in superspace into monogenic plane waves, which shall beuseful for inverting the super Radon transform.

Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace