MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS

By Amanda Folsom, Ken Ichi Ono, Robert C. Rhoades. Forum of Mathematics, Pi, 2013

89 citations recorded by the source corpus.

Abstract 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 $$\begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )\cdots (1- 2q+ 2{q}^{4} - \cdots )= O(1).\end{eqnarray*}$$ 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 $\vartheta $-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/19e3e5075ff51711a1083d581e3ac3eca19cc9c3

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Citations in the corpus

A q-microscope for supercongruences (2018, background): “We remark that asymptotic behavior of q-series at roots of unity attracts a lot of attention in recent studies of mock theta functions and so-called quantum modular forms; we limit our citations about related notion and results to [4, 33].”
Torus knots and quantum modular forms (2014, background): “(A.7) The dual U -functions, satisfying F (m) 3 (ζ −1 N ) = U (m) 3 (−1; ζN), are given by U (1) 3 (x; q) = ∞∑ n=0 n+1∑ k=1 k∑ j=1 (−xq)n (−x−1q)nqn−2+k 2+j2 [ k + j − 1 2j − 1 ] q [ n+ k + 2j − 1 2k + 2j − 2 ] q = 1 + q + (x+ 2 + x−1)q2 + (2x+ 4 + 2x−1)q3 + (4x+ 7 + 4x−1)q4 + · · · (A.8) U (2) 3 (x”