On Penrose integral formula and series expansion of kk-regular functions on the quaternionic space HnHn

The kk-Cauchy–Fueter operator can be viewed as the restriction to the quaternionic space HnHn of the holomorphic kk-Cauchy–Fueter operator on C4nC4n. A generalized Penrose integral formula gives the solutions to the holomorphic kk-Cauchy–Fueter equations, and conversely, any holomorphic solution to these equations is given by this integral formula. By restriction to the quaternionic space Hn⊆C4nHn⊆C4n, we find all kk-regular functions. The integral formula also gives the series expansion of a kk-regular function by homogeneous kk-regular polynomials. In particular, the result holds for left regular functions, which are exactly 11-regular. It is almost elementary to show the kk-regularity of the function given by the integral formula or such series, but the proof of the inverse part that any kk-regular function can be provided by the integral formula or such series involves some tools of sheaf theory.