In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients.