Recommendation systems have become ubiquitous tools in online platforms, providing personalized suggestions based on user-item interactions. To capture the dynamic higher-order connections between users and items, recommendation approaches based on dynamic graphs have garnered significant attention from researchers. However, existing recommendation methods based on dynamic graphs are often limited by data sparsity, which prevents them from achieving satisfactory performance. Fortunately, the rapid development of large language models (LLMs) with powerful text generation capabilities and extensive domain knowledge has offered new possibilities for addressing this challenge. However, how to effectively integrate LLMs with dynamic graphs remains unexplored. To bridge this gap, in this article, we propose a novel framework, that is, DynLLM, for applying LLMs to dynamic graph-based recommendation methods. Specifically, DynLLM harnesses the power of LLMs to generate multi-faceted user profiles based on the rich textual features of historical purchase records, which in turn supplement and enrich the underlying relationships between users and items. Along this line, to fuse the multi-faceted profiles with temporal graph embedding, we engage LLMs to derive corresponding profile embeddings and further employ a distilled attention mechanism to refine the LLM-generated profile embeddings for alleviating noisy signals, while also assessing and adjusting the relevance of each distilled facet embedding for seamless integration with temporal graph embedding from continuous time dynamic graphs (CTDGs). Extensive experiments on three real datasets have validated the superior improvements of DynLLM over a wide range of state-of-the-art baseline methods. The implementation code is available online at https://github.com/meteor-gif/DynLLM.