Formulas for Jacobi forms and generalized Frobenius partitions

TL;DR

This paper reformulates generating functions for generalized Frobenius partitions within the framework of Jacobi forms, enabling explicit formulas and recursive calculations that reveal their combinatorial structure and congruences.

Contribution

It introduces a novel approach by expressing Frobenius partition generating functions as Jacobi forms, facilitating explicit formulas and recursive computation methods.

Findings

Derived explicit formulas for generating functions

Established a recursion formula for calculations

Connected Frobenius partitions to Jacobi forms

Abstract

Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting combinatorial structure and led to proofs of a number of congruences. In this paper, we show how Andrews' generating functions can be cast in the framework of Eichler and Zagier's Jacobi forms. This reformulation allows us to compute explicit formulas for the generalized Frobenius partition generating functions (and in fact provides formulas for further functions of potential combinatorial interest), and it leads to a recursion formula to calculate them in terms of infinite $q$-products.