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The cache and transcoding of the multi-access edge computing (MEC) server and wireless resource allocation in eNodeB interact and determine the quality of experience (QoE) of dynamic adaptive streaming over HTTP (DASH) clients in MEC networks. However, the relationship among the three factors has not been explored, which has led to limited improvement in clients' QoE. Therefore, we propose a joint optimization framework of video segment cache and transcoding in MEC servers and resource allocation to improve the QoE of DASH clients. Based on the established framework, we develop a MEC cache management mechanism that consists of the MEC cache partition, video segment deletion, and MEC cache space transfer. Then, a joint optimization algorithm that combines video segment cache and transcoding in the MEC server and resource allocation is proposed. In the algorithm, the clients' channel state and the playback status and cooperation among MEC servers are employed to estimate the client's priority, video segment presentation switch and continuous playback time. Considering the above four factors, we develop a utility function model of clients' QoE. Then, we formulate a mixed-integer nonlinear programming mathematical model to maximize the total utility of DASH clients, where the video segment cache and transcoding strategy and resource allocation strategy are jointly optimized. To solve this problem, we propose a low-complexity heuristic algorithm that decomposes the original problem into multiple subproblems. The simulation results show that our proposed algorithms efficiently improve client's throughput, received video quality and hit ratio of video segments while decreasing the playback rebuffering time, video segment presentation switch and system backhaul traffic.