Generalized Frobenius partitions, Motzkin paths, and Jacobi forms

Abstract

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 \emph{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.