  In this paper we consider the classical $\bar{\partial}$-problem in the case
of one complex variable both for analytic and polyanalytic data. We apply the
decomposition property of polyanalytic functions in order to construct
particular solutions of this problem and obtain new H\"ormander type estimates
using suitable powers of the Cauchy-Riemann operator. We also compute
particular solutions of the $\bar{\partial}$-problem for specific polyanalytic
data such as the It\^o complex Hermite polynomials and polyanalytic Fock
kernels.
