Open-domain question answering (ODQA) has emerged as a pivotal research spotlight in information systems. Existing methods follow two main paradigms to collect evidence: (1) The \textit{retrieve-then-read} paradigm retrieves pertinent documents from an external corpus; and (2) the \textit{generate-then-read} paradigm employs large language models (LLMs) to generate relevant documents. However, neither can fully address multifaceted requirements for evidence. To this end, we propose LLMQA, a generalized framework that formulates the ODQA process into three basic steps: query expansion, document selection, and answer generation, combining the superiority of both retrieval-based and generation-based evidence. Since LLMs exhibit their excellent capabilities to accomplish various tasks, we instruct LLMs to play multiple roles as generators, rerankers, and evaluators within our framework, integrating them to collaborate in the ODQA process. Furthermore, we introduce a novel prompt optimization algorithm to refine role-playing prompts and steer LLMs to produce higher-quality evidence and answers. Extensive experimental results on widely used benchmarks (NQ, WebQ, and TriviaQA) demonstrate that LLMQA achieves the best performance in terms of both answer accuracy and evidence quality, showcasing its potential for advancing ODQA research and applications.