FALSE, PARTIAL, AND MOCK JACOBI THETA FUNCTIONS AS q-HYPERGEOMETRIC SERIES

to add 1. Curious q-series Identities Since Rogers [32] introduced the false theta functions, they have played a curious role in the theory of partitions (see for instance [1, 2, 12]). Andrews [3] defined a false theta function as any series of the shape ∑ n∈Z(±1)q 2+`n that has different signs for the nonzero terms. For example, Rogers proved (1.1) ∑ j≥0 q j(3j+1) 2 − ∑ j≤−1 q j(3j+1) 2 = ∑ j≥0 q j(3j+1) 2 (1− q) = ∞ ∑ n=0 (−1)nq n(n+1) 2 (1 + q)(1 + q2) · · · (1 + qn) . Andrews [3] proved this identity is equivalent to a near bijection between those partitions into distinct parts with odd largest part and even largest part. The unmodified series satisfy Jacobi’s triple product identity (1.2) ∑ n∈Z (±1)nqkn2+`n = ∞ ∏ n=1 (1− q)(1± q`−k+2kn)(1± q2kn−`−k). Such series are called theta functions and are, up to a power of q, a modular form. Andrews writes The fact that no results such as (1.2) holds for false theta functions surely diminishes their analytic interest. In fact, Ramanujan himself does not seem to have believed that the analytic theory of false theta functions was very natural. In his final letter to Hardy he wrote, I discovered very interesting functions recently which I call ‘Mock’ theta-functions. Unlike the ‘False’ theta-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as ordinary thetafunctions. We explain how certain classes of false theta functions, namely those that are partial theta functions, give rise to an analytic theory that is the same as that of the mock theta functions. The explanation uses the theory of mock Jacobi forms and provides additional evidence for the development of partial Jacobi forms. Simultaneously we discuss some curious q-series phenomenon and their explanations. Remark. The series in (1.1) may be rewritten as ∑ n≥0 q n(3n+1) 2 (1− q) = ∑ n≥0 ( −12 n ) q n2−1 24 , (1.3) Date: April 15, 2011. 2000 Mathematics Subject Classification. 11P55, 05A17.