With the popularity of Internet of Things (IoT), edge computing and cloud computing, more and more stream analytics applications are being developed including real-time trend prediction and object detection on top of IoT sensing data. One popular type of stream analytics is the recurrent neural network (RNN) deep learning model based time series or sequence data prediction and forecasting. Different from traditional analytics that assumes data are available ahead of time and will not change, stream analytics deals with data that are being generated continuously and data trend/distribution could change (a.k.a. concept drift), which will cause prediction/forecasting accuracy to drop over time. One other challenge is to find the best resource provisioning for stream analytics to achieve good overall latency. In this paper, we study how to best leverage edge and cloud resources to achieve better accuracy and latency for stream analytics using a type of RNN model called long short-term memory (LSTM). We propose a novel edge-cloud integrated framework for hybrid stream analytics that supports low latency inference on the edge and high capacity training on the cloud. To achieve flexible deployment, we study different approaches of deploying our hybrid learning framework including edge-centric, cloud-centric and edge-cloud integrated. Further, our hybrid learning framework can dynamically combine inference results from an LSTM model pre-trained based on historical data and another LSTM model re-trained periodically based on the most recent data. Using real-world and simulated stream datasets, our experiments show the proposed edge-cloud deployment is the best among all three deployment types in terms of latency. For accuracy, the experiments show our dynamic learning approach performs the best among all learning approaches for all three concept drift scenarios.