Bilateral series and Ramanujan’s radial limits

Bajpai, Jitendra; Kimport, Susie; Liang, Jie; Ma, Ding; Ricci, J.

doi:10.1090/s0002-9939-2014-12249-0

Cited by 18 publications

(17 citation statements)

References 8 publications

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“…. We obtain similar equations for the other cases of [3], Theorems 1.1. and 1.2. It would be interesting to find a direct explanation for all of these identities.…”

Section: Introductionsupporting

confidence: 64%

“…The problem has been solved for special mock theta functions before: Folsom, Ono, and Rhoades gave a closed expression for the radial limits of Ramanujan's third order mock theta function f (Theorem 1.1 of [8]) as well as a partial answer for Rhoades' question for certain specializations of g 3 (Theorem 1.2 of [8], see also Remark 2.4). Moreover, in [3], the radial limits of Ramanujan's fifth order mock theta functions were computed using bilateral series.…”

Section: Introductionmentioning

confidence: 99%

“…Let now k be a positive even integer with 5 ∤ k, so that (k, 10) = 2|2h for every h. Then we have h k ∈ Q 0,1,2,10 and we can apply Theorem 2.2. On the other hand, for even k, we can also apply the first equation of [3], Theorem 1.1. (a) to compute the radial limit.…”

Section: Introductionmentioning

confidence: 99%

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Radial limits of the universal mock theta function 𝑔₃

Jang

Löbrich

2016

Proc. Amer. Math. Soc.

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Referring to Ramanujan's original definition of a mock theta function, Rhoades asked for explicit formulas for radial limits of the universal mock theta functions g 2 and g 3 . Recently, Bringmann and Rolen found such formulas for specializations of g 2 . Here we treat the case of g 3 , generalizing radial limit results for the rank generating function of Folsom, Ono, and Rhoades. Furthermore, we find expressions for radial limits of fifth order mock theta functions different from those of Bajpai, Kimport, Liang, Ma, and Ricci.

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“…. We obtain similar equations for the other cases of [3], Theorems 1.1. and 1.2. It would be interesting to find a direct explanation for all of these identities.…”

Section: Introductionsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Radial limits of the universal mock theta function 𝑔₃

Jang

Löbrich

2016

Proc. Amer. Math. Soc.

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“…Remark: The results in Corollary 3.5 were also proved in [7], using somewhat similar arguments, as were the results in Corollary 4.2 below. The second implication is that they imply some summation formulae for some of the bilateral series appearing in Theorem 3.2.…”

Section: Mock Theta Functions Of the Third Ordersupporting

confidence: 55%

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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“…Here, g 3 (w; q) is the 'universal' mock theta function, aptly named, as it can be related to Ramanujan's original mock theta functions upon suitable specialization of its parameters (see [27]). See also recent related works by the first author [25], as well as [9], for more along these lines.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

Mock Theta Functions and Quantum Modular Forms

Folsom¹,

Ono²,

Rhoades³

2013

Forum math. Pi

114

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Ramanujan's last letter to Hardy concerns the asymptotic properties of modular forms and his 'mock theta functions'. For the mock theta function f (q), Ramanujan claims that as q approaches an even-order 2k root of unity, we have f (q) − (−1) k (1 − q)(1 − q 3 )(1 − q 5 ) · · · (1 − 2q + 2q 4 − · · ·) = O(1).We prove Ramanujan's claim as a special case of a more general result. The implied constants in Ramanujan's claim are not mysterious. They arise in Zagier's theory of 'quantum modular forms'. We provide explicit closed expressions for these 'radial limits' as values of a 'quantum' q-hypergeometric function which underlies a new relationship between Dyson's rank mock theta function and the Andrews-Garvan crank modular form. Along these lines, we show that the Rogers-Fine false ϑ-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms. 2010 Mathematics Subject Classification: 11F99 (primary); 11F37, 33D15 (secondary) Overview In his 1920 deathbed letter to Hardy, Ramanujan gave examples of 17 curious q-series he referred to as 'mock theta functions' [11]. In the decades following Ramanujan's death, mathematicians were unable to determine how these functions fit into the theory of modular forms, despite c The Author(s) 2013. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence . A. Folsom, K. Ono, and R. C. Rhoades 2 their rather ubiquitous nature. Finally, the 2002 Ph.D. thesis of Zwegers [45] showed that while the mock theta functions were not modular, they could be 'completed' to produce real analytic vector-valued modular forms. Zwegers's breakthrough catalyzed the development of the overarching theory of 'weak Maass forms' by Bringmann, Ono, and collaborators [15, 16, 35, 42]. Ramanujan's mock theta functions, it turns out, are examples of 'holomorphic parts' of weak Maass forms, originally defined by Bruinier and Funke [18].While the theory of weak Maass forms has led to a flood of applications in many disparate areas of mathematics (see [35,42] and references therein), it is still not the case that we fully understand the deeper framework surrounding the contents of Ramanujan's last letter to Hardy. Here, we revisit Ramanujan's original claims and motivations. His last letter summarizes asymptotic properties near roots of unity of modular 'Eulerian' series. Ramanujan asks whether other Eulerian series with similar asymptotics are necessarily the sum of a modular theta function and a function which is O(1) at all roots of unity. He writes: 'The answer is it is not necessarily so . . . I have not proved rigorously that it is not necessarily so . . . But I have constructed a number of examples . . . '. In fact, Ramanujan's sole example and claim pertain to his third-order mock theta function f (q). CLAIM (Ramanujan [11]). As q approaches an even-order 2k root of unity, we haveHere, we prove (in ...

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Bilateral series and Ramanujan’s radial limits

Cited by 18 publications

References 8 publications

Radial limits of the universal mock theta function 𝑔₃

Radial limits of the universal mock theta function 𝑔₃

Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

Mock Theta Functions and Quantum Modular Forms

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