Cauchy and Poisson formulas for polyanalytic functions and applications

We obtain new Cauchy and Poisson integral formulas for polyanalytic functions. As an application, we establish mean value theorems for functions polyanalytic and real polyharmonic in a disk. We also give applications to sharp estimates of generalized maximum modulus principle type for associated functions, and, in particular, to estimates for rational functions (components) in the problem of singularity separation for polyrational functions.