This paper studies fog computing systems, in which cloud data centers can be supplemented by a large number of fog nodes deployed in a wide geographical area. Each node relies on harvested energy from the surrounding environment to provide computational services to local users. We propose the concept of dynamic network slicing in which a regional orchestrator coordinates workload distribution among local fog nodes, providing partitions/slices of energy and computational resources to support a specific type of service with certain quality-of-service (QoS) guarantees. The resources allocated to each slice can be dynamically adjusted according to service demands and energy availability. A stochastic overlapping coalition-formation game is developed to investigate distributed cooperation and joint network slicing between fog nodes under randomly fluctuating energy harvesting and workload arrival processes. We observe that the overall processing capacity of the fog computing network can be improved by allowing fog nodes to maintain a belief function about the unknown state and the private information of other nodes. An algorithm based on a belief-state partially observable Markov decision process (B-POMDP) is proposed to achieve the optimal resource slicing structure among all fog nodes. We describe how to implement our proposed dynamic network slicing within the 3GPP network sharing architecture, and evaluate the performance of our proposed framework using the real BS location data of a real cellular system with over 200 BSs deployed in the city of Dublin. Our numerical results show that our framework can significantly improve the workload processing capability of fog computing networks. In particular, even when each fog node can coordinate only with its closest neighbor, the total amount of workload processed by fog nodes can be almost doubled under certain scenarios.