Polynomial sequences generated by integral powers of first and second order differential operators with polynomial coefficients will be under discussion. More precisely, the focus will lie on their connection with well known orthogonal polynomial sequences along with their foremost structural properties. We will start by analysing the cases in which the aforementioned differential operator is of first order, bringing into analysis polynomial sequences associated to the classical linear functionals of Hermite, Laguerre, Bessel and Jacobi. Afterwards, the discussion will proceed towards the analysis of polynomial sequences generated by second order differential operators, which brings up the open problem of characterizing  orthogonal polynomial sequences with respect to certain positive definite linear functionals. The Kontorovich-Lebedev transform and the central factorials will be an asset to attain our goals.