Congruences for Taylor expansions of quantum modular forms

Guerzhoy, Pavel; Kent, Zachary A.; Rolen, Larry

doi:10.1186/s40687-014-0017-2

Cited by 9 publications

(7 citation statements)

References 38 publications

(57 reference statements)

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“…Congruences for the coefficients of the functions F (q) and G(q) in Theorems 1.2 and 1.3 can be deduced from the results of [7]. In closing we mention another approach.…”

Section: Remarks On Congruencesmentioning

confidence: 93%

“…2) In subsequent work of the first two authors, Garvan, and Straub [1,6,12], similar congruences were obtained for prime powers and for generalized Fishburn numbers. Taking a different approach, Guerzhoy, Kent, and Rolen [7] interpreted the coefficients in the asymptotic expansions of functions P (1) a,b,χ (e −t ) defined in (1.8) below in terms of special values of L-functions, and proved congruences for these coefficients using divisibility properties of binomial coefficients. These congruences are inherited by any function whose expansion at q = 1 agrees with one of these expansions; these include the function F (q) and, more generally, the Kontsevich-Zagier functions described in Section 5 below.…”

Section: Introductionmentioning

confidence: 99%

“…These congruences are inherited by any function whose expansion at q = 1 agrees with one of these expansions; these include the function F (q) and, more generally, the Kontsevich-Zagier functions described in Section 5 below. See [7] for details.…”

Section: Introductionmentioning

confidence: 99%

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Dissections of Strange q-Series

2019

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In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich-Zagier series are divisible by a certain q-factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of q-hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.

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“…Congruences for the coefficients of the functions F (q) and G(q) in Theorems 1.2 and 1.3 can be deduced from the results of [7]. In closing we mention another approach.…”

Section: Remarks On Congruencesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Dissections of Strange q-Series

2019

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“…This result, it turns out, is not limited to the mock theta function f , and has been generalized to other mock modular forms, including the twovariable rank generating function R, in [36,69,101]. We do not give a full treatment of the developing subject of quantum modular forms here, but note that there has been much recent progress in the area; in addition to the references mentioned above, the interested reader may also wish to consult, for example, [24,26,37,94,97,105,125,140].…”

Section: Ramanujan's Mock Theta Functionsmentioning

confidence: 99%

Perspectives on mock modular forms

Folsom

2017

Journal of Number Theory

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Mock modular forms have played many prominent roles in number theory and other areas of mathematics over the course of the last 15 years. While the term "mock modular form" was not formally defined in the literature until 2007, we now know in hindsight that evidence of this young subject appears much earlier, and that mock modular forms are intimately related to ordinary modular and Maass forms, and Ramanujan's mock theta functions. In this expository article, we offer several different perspectives on mock modular forms-some of which are number theoretic and some of which are not-which together exhibit the strength and scope of their developing theory. They are: combinatorics, q-series and mock theta functions, mathematical physics, number theory, and Moonshine. We also describe some essential results of Bruinier and Funke, and Zwegers, both of which have made tremendous impacts on the development of the theory of mock modular forms. We hope that this article is of interest to both number theorists and enthusiasts-to any reader who is interested in or curious about the history, development, and applications of the subject of mock modular forms, as well as some amount of the mathematical details that go along with them. 1 Preamble Modular forms have played central and far-reaching roles in number theory and mathematics over the last two centuries. Their footprints can be seen in the elliptic functions of the early 1800s, and their prominence has only risen since then. The theory and importance of their younger descendants, mock modular forms, has evolved in many analogous ways. Among the many papers written in the subject of mock modular forms are three excellent articles which are accessible to the non-specialist, and which we recommend to the readers of this article: Duke's 2014 article [54] in the the Notices of the AMS, Ono's 2010 article [120] in the Notices of the AMS, and Zagier's 2007 Séminaire Bourbaki article [138]. These articles highlight different aspects of the history, theory, and applications of mock modular forms, including discussions of Ramanujan's mock theta functions from 1920. The general term mock modular form was not defined in the literature until 2007 [138], and we now know in hindsight that Ramanujan's mock theta functions are among the oldest examples. In this expository article, we begin by offering a few concrete examples of mock modular forms in Section 2, which can be viewed as companions to some familiar ordinary modular forms. We offer these examples as previews to the more formal definitions and basic properties given in Section 3. In all of the following sections, we offer several perspectives on mock modular forms from the standpoint of different areas in which mock modular forms 1

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“…Here, the moniker "strange" is used as F (q) does not converge on any open subset of C, but is well-defined when q is a root of unity (where it is finite) and when q is replaced by 1 − q as in (1.1). The Fishburn numbers are of interest for their numerous combinatorial variants (see A022493 in [23]), asymptotics [24,28] and arithmetic properties [1,3,7,8,25]. In their marvelous paper, Andrews and Sellers [3] proved congruences for ξ(n) modulo primes which were then extended to prime powers [1,25].…”

Section: Introductionmentioning

confidence: 99%

Generalized Fishburn numbers and torus knots

Bijaoui,

Boden,

Myers

et al. 2020

Preprint

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Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the 1 − q expansion of the Kontsevich-Zagier series Ft(q) for the torus knots T (3, 2 t ), t ≥ 2. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new "strange identity" for Ft(q).

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Congruences for Taylor expansions of quantum modular forms

Cited by 9 publications

References 38 publications

Dissections of Strange q-Series

Dissections of Strange q-Series

Perspectives on mock modular forms

Generalized Fishburn numbers and torus knots

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