HTTP based adaptive video streaming has become a popular choice of streaming due to the reliable transmission and the flexibility offered to adapt to varying network conditions. However, due to rate adaptation in adaptive streaming, the quality of the videos at the client keeps varying with time depending on the end-to-end network conditions. Further, varying network conditions can lead to the video client running out of playback content resulting in rebuffering events. These factors affect the user satisfaction and cause degradation of the user quality of experience (QoE). It is important to quantify the perceptual QoE of the streaming video users and monitor the same in a continuous manner so that the QoE degradation can be minimized. However, the continuous evaluation of QoE is challenging as it is determined by complex dynamic interactions among the QoE influencing factors. Towards this end, we present LSTM-QoE, a recurrent neural network based QoE prediction model using a Long Short-Term Memory (LSTM) network. The LSTM-QoE is a network of cascaded LSTM blocks to capture the nonlinearities and the complex temporal dependencies involved in the time varying QoE. Based on an evaluation over several publicly available continuous QoE databases, we demonstrate that the LSTM-QoE has the capability to model the QoE dynamics effectively. We compare the proposed model with the state-of-the-art QoE prediction models and show that it provides superior performance across these databases. Further, we discuss the state space perspective for the LSTM-QoE and show the efficacy of the state space modeling approaches for QoE prediction.