Ramanujan's Lost Notebook

Abstract

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An extension of the Andrews–Warnaar partial theta function identity

“…(aq n+1 ; q) n q n (q; q) n (aq 2 ; q 2 ) n = 1 (q; q) ∞ (aq 2 ; q 2 ) ∞ ∞ n=0 (−1) n a n q n(n+1) , which was also given by Warnaar [32, (5.3)]. The above identity is the residual identity corresponding to Ramanujan's partial theta function identity [7,Entry 6.3.11].…”

“…As a first application, we derive a two-term partial theta function identity from (3.1). We begin with an example from Ramanujan's Lost Notebook [27, p.28], see also Entry 1.6.2 in [7].…”

APS η-invariant, path integrals, and mock modularity

“…We obtained the quantum modular form as a limit of the completion of a mock modular form defined on the upper half τ plane H + (τ 2 > 0) but it can equally well be obtained as a limit of a false theta function defined on the lower half plane H − (τ 2 < 0). Consider the false theta function [58,59] defined by With r = + 2kn, one can rewrite the f (τ 1 ) as…”

On the spectrum of a partial theta function

“…This terminology is explained by the resemblance of formula (1) with the one defining the Jacobi theta function Θ(q, z) := ∞ j=−∞ q j 2 z j ; the word "partial" refers to the summation in the case of θ taking place only over the nonnegative values of j. One has θ(q 2 , z/q) = ∞ j=0 q j 2 z j .…”

“…Various analytic properties of θ are studied in [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and [21]. See more about θ in [1].…”