Modular forms and quantum invariants of 3-manifolds

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“…We see that the modular property of Ψ (a) P (τ ), especially the modular S-transformation (2.3), leads a nearly modular property of the Eichler integral as [29] (see also Refs. 19, 46)…”

“…29, definitions (3.1) and (3.2) can be extended to outside the unit circle |q| > 1, and we define new functions by substituting 1/q in place of q;…”

“…Nearly modular property has also revealed in recent studies of the quantum invariant [29]. Therein the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant τ N (M) [38,43] for the Poincaré homology sphere M = Σ(2, 3, 5) was identified with a limiting value of the Eichler integral of the modular form with half-integral weight in τ → 1/N, and an exact asymptotic expansion in N → ∞ was given by use of the modular transformation.…”

“…The Eichler integral of this set of the modular forms with half-integral weight is then defined as a half-integration of Ψ (a) P (τ ) with respect to τ by [19,29] …”

Radial limits of partial theta and similar series

“…(10.5)]. Coefficients in the asymptotic expansions of both (1) and (2) are explicitly given in [8]. In particular, lim q→1 − n≥0…”