scholarly journals The generating function of the $M_2$-rank of partitions without repeated odd parts as a mock modular form

2018 ◽  
Vol 371 (1) ◽  
pp. 249-277 ◽  
Author(s):  
Chris Jennings-Shaffer
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Mock Modular Form

scholarly journals A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL M. KANE ◽  
ROBERT C. RHOADES
Keyword(s):  
Asymptotic Behavior ◽  
Modular Form ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Bootstrap Percolation ◽  
Asymptotic Result ◽  
Integer Partitions ◽  
Mock Modular Form ◽  
Precise Asymptotic

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

scholarly journals Overpartitions with Restricted Odd Differences

10.37236/5248 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jehanne Dousse ◽  
Jeremy Lovejoy ◽  
Karl Mahlburg
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Difference Equations ◽  
Circle Method ◽  
Hecke Operators ◽  
Asymptotic Formulas ◽  
Linear Congruences ◽  
The Difference ◽  
Mock Modular Form

We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.

scholarly journals Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Taylor Garnowski
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Asymptotic Distribution ◽  
Generating Function ◽  
Modular Forms ◽  
Asymptotic Estimate ◽  
Asymptotic Properties ◽  
Unimodal Sequences ◽  
The World ◽  
Mock Modular Form

AbstractKim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to $$a \pmod {c}$$ a ( mod c ) for $$c\ne 2$$ c ≠ 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

scholarly journals Dissections of a "strange" function

2015 ◽  
Vol 11 (05) ◽  
pp. 1557-1562 ◽  
Author(s):  
Scott Ahlgren ◽  
Byungchan Kim
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Generating Function ◽  
Prime Power ◽  
Partial Sums ◽  
Root Of Unity ◽  
Prime Modulus

The "strange" function of Kontsevich and Zagier is defined by [Formula: see text] This series is defined only when q is a root of unity, and provides an example of what Zagier has called a "quantum modular form". In their recent work on congruences for the Fishburn numbers ξ(n) (whose generating function is F(1 - q)), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of F(q). We prove that a more general form of their speculation is true. The congruences of Andrews–Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for ξ(n) modulo prime powers.

scholarly journals On the Modular Completion of Certain Generating Functions

2020 ◽  
Author(s):  
Kathrin Bringmann ◽  
Stephan Ehlen ◽  
Markus Schwagenscheidt
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Modular Forms ◽  
Generating Functions ◽  
Natural Family ◽  
Weakly Holomorphic Modular Forms ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Generating Series ◽  
Holomorphic Modular Forms

Abstract We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [ 27] on traces of singular moduli.

scholarly journals Black holes and Hurwitz class numbers

2017 ◽  
Vol 26 (12) ◽  
pp. 1742003 ◽  
Author(s):  
Shamit Kachru ◽  
Arnav Tripathy
Keyword(s):  
Black Holes ◽  
Number Theory ◽  
String Theory ◽  
Modular Form ◽  
Counting Function ◽  
Class Numbers ◽  
Type Ii ◽  
Mock Modular Form

We define a natural counting function for BPS black holes in [Formula: see text] compactification of type II string theory, and observe that it is given by a weight 3/2 mock modular form discovered by Zagier. This hints at tantalizing relations connecting black holes, string theory, and number theory.

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