We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.

中文翻译：

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Fine 分区定理的新改进

我们发现了 Fine 的分区定理的新改进，将分区划分为具有最小部分奇数的不同部分。结果，我们获得了两个伴随分区标识。提供了分析和组合证明。