Sixth order mock theta functions

Berndt, Bruce C.; Chan, Ssc

doi:10.1016/j.aim.2007.06.004

Cited by 42 publications

(33 citation statements)

References 8 publications

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“…Between the time of Ramanujan's death in 1920 and the early part of the 21st century, approximately 35 other q-series were studied and deemed mock theta functions. Some were introduced by Watson [42], some were found in Ramanujan's lost notebook and studied by Andrews, Choi, and Hickerson [7,20,21,22,23,24], and others were produced by Berndt, Chan, Gordon and McIntosh using intuition from q-series [10,30,34]. For a summary of this classical work, see [31] or [32].…”

Section: Introductionmentioning

confidence: 99%

The Bailey chain and mock theta functions

Lovejoy

Osburn

2013

Advances in Mathematics

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Standard applications of the Bailey chain preserve mixed mock modularity but not mock modularity. After illustrating this with some examples, we show how to use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of q-hypergeometric multisums which are mock theta functions. We also prove identities involving some of these multisums and certain classical mock theta functions.

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Section: Introductionmentioning

confidence: 99%

The Bailey chain and mock theta functions

Lovejoy

Osburn

2013

Advances in Mathematics

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“…The following representations for sixth order mock theta functions were proved by Andrews and Hickerson [8], and Berndt and Chan [10].…”

Section: Mock Theta Functions Of Ordermentioning

confidence: 88%

“…In Ramanujan's original definition of µ (6) (q), the series does not converge, and (7.6) is the correct understanding of his definition. In 2007, Berndt and Chan [10] defined two new mock theta functions as:…”

Section: Mock Theta Functions Of Ordermentioning

confidence: 99%

Representations of mock theta functions

Chen

Wang

2020

Advances in Mathematics

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Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters a and b. Specializing the choices of (a, b), we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same (a, b)-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks m(x, q, z) and f a,b,c (x, y, q) introduced by Hickerson and Mortenson, as well as m(x, q, z) and f a,b,c (x, y, q) introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.

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“…Andrews and Hickerson [5] proved a number of identities for the sixth order mock theta functions stated by Ramanujan in the Lost Notebook [23]. Berndt and Chan [8] proved a number of similar identities. The proofs in both of these papers were quite involved, employing both Bailey pairs and the constant term method, and simpler proofs were later given by Lovejoy [19], for four of the identities proved by Andrews and Hickerson [5].…”

Section: Mock Theta Functions Of the Sixth Ordermentioning

confidence: 99%

Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

Laughlin

2017

Springer Proceedings in Mathematics &Amp; Statistics

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This paper is dedicated to Krishna Alladi on the occasion of his 60 th birthday.Abstract The bilateral series corresponding to many of the third-, fifth-, sixth-and eighth order mock theta functions may be derived as special cases of 2 ψ 2 seriesThree transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say χ(q), and a root of unity in a certain class, say ζ , that there is a theta function θ χ (q) such that limexists, as q → ζ from within the unit circle.where the first series is g 2 (x, q), the universal mock theta function of Gordon and McIntosh [17, Eq. (4.11)], g 3 (x; q) is as defined at (3.1), and j(x; q) := (x, q/x, q; q) ∞ , J a,m := j(q a ; q m ), J a,m := j(−q a ; q m ), J m := J m,3m = (q m ; q m ) ∞ . As Mortenson indicated in [22], if a mock theta function is expressible in terms of g 2 (x, q) and combinations of infinite products, then it may be possible to derive a n (−ζ 8 ; ζ 8 ) n = lim q→ζ J 1,2 J 6,16 J 2,8 − 2qJ 3 16 J 8,16 J 2,16 − J 10 16J 2,16 J 4 8 J 4 32 J 2,16J10,16, (6.20) and once again, experiment seems to suggest that each side is identically zero when ζ is any primitive root of unity of odd order. As with the hypotheses suggested by experiment for mock theta functions of sixth order, we have not attempted to prove these assertions. Results similar to those described above may be derived for other eighth order mock theta functions.

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Sixth order mock theta functions

Cited by 42 publications

References 8 publications

The Bailey chain and mock theta functions

The Bailey chain and mock theta functions

Representations of mock theta functions

Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

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