The fifth generation (5G) of mobile communications systems plans to support different types of applications, where each of these applications may have different requirements. For this reason, the main characteristic of 5G mobile communications systems is the flexibility of their architecture. An interesting proposal that can meet the requirements of these applications is the multi-carrier waveform Generalized Frequency Division Multiplexing (GFDM). However, due to its multi-carrier nature, this new waveform is highly sensitive to non-linear distortions arising mainly from high-power amplifiers (HPA). In addition, the wideband characteristics of multi-carrier signals result in frequency-dependent distortions, typically know as memory effects. This paper presents the development of closed-form analytical expressions that could be used to evaluate the impact of the distortions induced by non-linearities with memory in the out-of-band emissions and the bit error rate performance of GFDM-based systems. The resulting analytical expressions are general enough to obtain numerical results for different parameters of the GFDM-based system.