Robert C. Rhoades

Eulerian series as modular forms

In 1988, Hickerson proved the celebrated “mock theta conjectures”, a collection of ten identities from Ramanujan’s “lost notebook” which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

Unimodal sequences and quantum and mock modular forms

We show that the rank generating function U(t; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U(-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F(q). As a result, we obtain a new representation for a certain generating function for L-values. The series U(i; q) = U(-i; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.

Asymptotics for rank and crank moments

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture of two of the authors that refined a conjecture of Garvan. Garvans original conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The refined version provides precise asymptotic estimates for both the moments and their differences. Our proof uses the Hardy–Ramanujan Circle Method, multiple sums of Bernoulli polynomials and the theory of quasimock theta functions.

Taylor 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 coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms. As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

Duality and Differential Operators for Harmonic Maass Forms

Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator \({D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}\) acting on a harmonic Maass form for integers k > 2 in terms of \({\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}\) acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the p-adic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms.

On Ramanujan’s definition of mock theta function

In his famous “deathbed” letter, Ramanujan “defined” the notion of a mock theta function and offered some examples of functions he believed satisfied his definition. Very recently, Griffin et al. established for the first time that Ramanujan’s mock theta functions actually satisfy his own definition. On the other hand, Zwegers’ 2002 doctoral thesis [Zwegers S (2002) Mock theta functions. PhD thesis (Univ Utrecht, Utrecht, The Netherlands)] showed that all of Ramanujan’s examples are holomorphic parts of harmonic Maass forms. This has led to an alternate definition of a mock theta function. This paper shows that Ramanujan’s definition of mock theta function is not equivalent to the modern definition.

Central limit theorems for some set partition statistics

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

Closed expressions for averages of set partition statistics

In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field, we found that the moments (mean, variance, and higher moments) of novel statistics on set partitions of [n]={1,2,⋯,n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.

Note on a partition limit theorem for rank and crank

If lambda is a partition of n, then the rank rk(lambda) is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, in K. Bringmann, K. Mahlburg, and R. C. Rhoades (Bull. Lond. Math. Soc., 43 (2011) 661-672), it is shown that has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of beta(T), where beta(t) is standard Brownian motion and T is the first time that an independendent 3-dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.

Asymptotics for the number of row-Fishburn matrices

In this paper, we provide an asymptotic for the number of row-Fishburn matrices of size n which settles a conjecture by Vit Jelinek. Additionally, using q-series constructions we provide new identities for the generating functions for the number of such matrices, one of which was conjectured by Peter Bala.