Derivatives and Eulerian Numbers

The aim of the present note is to give a closed form formula for the nth derivative of a function which satisfies Riccati’s differential equation with constant coefficients. The paper is organized as follows. In section 2 we recall the definition and basic properties of Eulerian numbers. We introduce and prove the mentioned formula in section 3. Some examples where the formula can be applied are included in section 4. In one of them, using a formula derived by Hamilton, we arrive at an elegant integral expression for Eulerian numbers.