Rank-crank Type Pde’s and Non-holomorphic Jacobi Forms
نویسنده
چکیده
In this paper we show how Rank-Crank type PDE’s (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.
منابع مشابه
Mock Jacobi Forms in Basic Hypergeometric Series
We show that some q-series such as universal mock theta functions are linear sums of theta quotient and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are multiplied by suitable powers of q. And we prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or com...
متن کاملRank-crank Type Pde’s for Higher Level Appell Functions
In this paper we consider a level l Appell function, and find, for all odd l, a partial differential equation it satisfies. For l = 3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l = 5 we get a similar PDE found by Garvan.
متن کاملTaylor Coefficients of Mock-jacobi Forms and Moments of Partition Statistics
We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coeff...
متن کاملQuantum Jacobi forms and finite evaluations of unimodal rank generating functions
In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expre...
متن کاملExact Formulas for Coefficients of Jacobi Forms
In previous work, we introduced harmonic Maass-Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass-Jacobi-Poincaré series, as well as Zwegers’ real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass-Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this...
متن کامل