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Quantum q-series identities

Abstract : As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.
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https://hal.archives-ouvertes.fr/hal-03498183
Contributor : Srinivas Kotyada Connect in order to contact the contributor
Submitted on : Monday, December 20, 2021 - 9:14:03 PM
Last modification on : Tuesday, March 29, 2022 - 4:01:35 AM

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Jeremy Lovejoy. Quantum q-series identities. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2022, Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.61 -- 73. ⟨10.46298/hrj.2022.8930⟩. ⟨hal-03498183⟩

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