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As a powerful tool for modeling complex relationships, hypergraphs are gaining popularity from the graph learning community. However, commonly used frameworks in deep hypergraph learning focus on hypergraphs with edge-independent vertex weights (EIVWs), without considering hypergraphs with edge-dependent vertex weights (EDVWs) that have more modeling power. To compensate for this, we present General Hypergraph Spectral Convolution (GHSC), a general learning framework that not only handles EDVW and EIVW hypergraphs, but more importantly, enables theoretically explicitly utilizing the existing powerful Graph Convolutional Neural Networks (GCNNs) such that largely ease the design of Hypergraph Neural Networks. In this framework, the graph Laplacian of the given undirected GCNNs is replaced with a unified hypergraph Laplacian that incorporates vertex weight information from a random walk perspective by equating our defined generalized hypergraphs with simple undirected graphs. Extensive experiments from various domains including social network analysis, visual objective classification, and protein learning demonstrate the state-of-the-art performance of the proposed framework.