We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite q-products. In particular, we show that specific examples of our result easily reestablish previously known formulas, and we describe new congruences, both conjectural and proven, in additional cases. The modular structure of Jacobi forms indicates that all of the coefficients of the forms are of interest. We give a combinatorial definition of these “companion series” and explore their combinatorics via the counting of Motzkin paths.

我们展示了如何在 Eichler 和 Zagier 的理论中将广义 Frobenius 分区的 Andrews 生成函数理解为某些 Jacobi 形式的特定系数。这种重新表述导致一个递归过程，该过程根据无限q积产生广义 Frobenius 分区生成函数的显式公式。特别是，我们展示了我们结果的特定示例很容易重新建立以前已知的公式，并且我们在其他情况下描述了推测和证明的新同余。Jacobi 形式的模结构表明形式的所有系数都是感兴趣的。我们给出了这些“同伴系列”的组合定义，并通过对 Motzkin 路径的计数来探索它们的组合。