Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

We prove that the homogeneously polyanalytic functions of total order m, defined by system equations $$\overline{D}^{(k_1,\ldots ,k_n)} f=0$$ with $$k_1+\cdots +k_n=m$$ , can be written as polynomials degree $$<m$$ in variables $$\overline{z_1},\ldots ,\overline{z_n}$$ some analytic coefficients. establish a weighted mean value property for such functions, using reproducing Jacobi polynomials. After that, we give general recipe to transform kernel change variables. Applying these tools, compute Bergman space on unit ball $$\mathbb {C}^n$$ and Siegel domain. For one-dimensional case, analogous results were obtained Koshelev (Akad. Nauk SSSR 232: 277–279, 1977), Pessoa (Complex Anal. Oper. Theory 8: 359–381, 2014), Hachadi Youssfi 13: 3457–3478, 2019).