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Frank Garvan

Publications and source records attributed to Frank Garvan.

At least 19 recordsLinked to original sources
arXiv2026

Some convolution identities for mock modular forms arising from the theory of holomorphic projection

Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamo\u{g}lu, Raum, and Richter from 2014. Imamo\u{g}lu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamo\u{g}lu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.

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arXiv2026

Sum of parts in overpartitions and partitions without repeated odd parts

In this paper, we obtain several Ramanujan-type congruences modulo 5 and 7 for sum of certain non-overlined parts in overpartitions classified by parity and sum of certain parts in partitions without repeated odd parts classified by parity. Our proofs for the congruences are elementary, depending only on classical theta function identities.

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arXiv2026

2- and 3-Dissections of Second-, Sixth-, and Eighth-Order Mock Theta Functions

In this paper, we develop a systematic method for obtaining and proving $m$-dissections of mock theta functions. In 2014, Hickerson and Mortenson showed how to derive and prove identities for Ramanujan's mock theta functions and Hecke-type indefinite theta series using Appell--Lerch sums. We build on their transformation formula method, combining it with symbolic computations and algorithms for the theory of modular functions. We focus exclusively on the cases of 2- and 3-dissections.

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arXiv2025

Transformation of Third Order Mock Theta Functions and New $q$-Series Identities

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions $f(q)$ and $\omega$(q). Zwegers in 2000 built on Watson's techniques to complete these mock theta functions and connected them to real analytic modular forms. We show how to derive these transformations using Lerch sums. To show the equivalence of the results involves some new $q$-series identities thus resulting in a new proof of Zwegers' theorem.

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arXiv2025

An infinite family of overpartition congruences mod powers of 2

We prove an infinite family of Hecke-like congruences for the overpartition function modulo powers of 2. Starting from a recent identity of Garvan and Morrow and iterating Atkin's $U_2$ operator, we determine lower bounds on the 2-adic valuations of the coefficients that arise at each step. Our approach yields new modular equations relating the Hauptmoduln $G_2$ on $\Gamma_0(2)$ and $G_8$ on $\Gamma_0(8)$, together with explicit $U_2$-action formulas.

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arXiv2025

Multiplicative congruences for Andrews's even parts below odd parts function and related infinite products

We prove multiplicative congruences mod $2^{12}$ for George Andrews's partition function, $\overline{\mathcal{EO}}(n)$, the number of partitions of $n$ in which every even part is less than each odd part and only the largest even part occurs an odd number of times. We find analogous congruences for more general infinite products. These congruences are obtained using Fricke involutions and Newman's approach to half integer weight Hecke operators on eta quotients, and were inspired by Atkin's multiplicative congruences for the partition function.

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arXiv2024

Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions

It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of $5$ for the crank parity function. The generating function for the analogous rank parity function is $f(q)$, the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of $5$ for the rank parity function, and here we extend these congruences for powers of $7$. We also show how these congruences imply congruences modulo powers of $5$ and $7$ for the coefficients of the related third order mock theta function $\omega(q)$, using Atkin-Lehner involutions and transformation results of Zwegers. Finally we a prove a family of congruences modulo powers of $7$ for the crank parity function.

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arXiv2021

Congruences modulo powers of 5 for the rank parity function

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

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arXiv2020

A new approach to the Dyson rank conjectures

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of Ramanujan's famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of 7n+5. In 1954 Atkin and Swinnerton-Dyer proved Dyson's rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke-Rogers series.

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arXiv2020

A proof of the mod $4$ unimodal sequence conjectures and related mock theta functions

In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4 congruences for the coefficients of these generating functions. In 2016 Kim, Lim and Lovejoy obtained similar results for odd-balanced unimodal sequences and made similar mod 4 conjectures. We prove all of these mod 4 conjectures and similar congruences for the Andrews spt-function and related mock theta functions. Our method of proof involves new Hecke-Rogers type identities for indefinite binary quadratic forms and the Hurwitz class number.

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arXiv2019

New fifth and seventh order mock theta function identities

We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions $χ_0(q)$ and $χ_1(q)$ and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

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arXiv2019

A tutorial for the MAPLE ETA package

This is a tutorial for using ETA, a MAPLE package for calculating with Dedekind's eta function. The ETA package is designed for proving eta-product identities using the valence formula for modular functions.

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arXiv2018

Automatic Proof of Theta-Function Identities

This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan's 40 identities for his so called Rogers-Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for generalizations of Ramanujan's G(q) and H(q) and Robin's extensions. These generalizations are associated with certain real Dirichlet characters. We find a total of over 150 identities.

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arXiv2016

Transformation properties for Dyson's rank function

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(ζ,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $ζ$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(ζ,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

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arXiv2016

Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E

We investigate spt-crank-type functions arising from Bailey pairs. We recall four spt-type functions corresponding to the Bailey pairs $A1$, $A3$, $A5$, and $A7$ of Slater and given four new spt-type functions corresponding to Bailey pairs $C1$, $C5$, $E2$, and $E4$. Each of these functions can be thought of as a count on the number of appearances of the smallest part in certain integer partitions. We prove simple Ramanujan type congruences for these functions that are explained by a spt-crank-type function. The spt-crank-type functions are two variable $q$-series determined by a Bailey pair, that when $z=1$ reduce to the spt-type functions. We find the spt-crank-type functions to have interesting representations as either infinite products or as Hecke-Rogers-type double series. These series reduce nicely when $z$ is a certain root of unity and allow us to deduce the congruences. Additionally we find dissections when $z$ is a certain root of unity to give another proof of the congruences. Our double sum and product formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.

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arXiv2016

Combinatorial interpretations of Ramanujan's tau function

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.

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arXiv2014

The spt-crank for overpartitions

Bringmann, Lovejoy, and Osburn showed that the generating functions of the spt-overpartition functions spt(n), spt1(n), spt2(n), and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences mod 5 and 7 for spt (n). Chen, Ji, and Zang were able to define this spt-crank in terms of ordinary partitions. In this paper we define spt-cranks in terms of vector partitions that combinatorially explain the known simple congruences for all the spt-overpartition functions as well as new simple congruences. For all the overpartition functions except M2spt(n) we are able to define the spt-crank purely in terms of marked overpartitions. The proofs of the congruences depend on Bailey's Lemma and the difference formulas for the Dyson rank of an overpartition and the M2-rank of a partition without repeated odd parts.

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