Abstract

Ramanujan introduced his now celebrated mock theta functions in 1920, grouping them into families parameterized by an integer called the order. In 2010 Bringmann and Ono discovered generalizations of Ramanujan's mock theta functions for any order relatively prime to 6; this result was later strengthened by Garvan in 2016. It was also shown that by adding suitable nonholomorphic completion terms to the mock theta functions the family of mock theta functions corresponding to a given order constitute a complex vector space which is closed under the action of the modular group. We strengthen the Bringmann, Ono, and Garvan result by constructing a vector-valued modular form of weight 1/2 transforming according the Weil representation for orders greater than 3 by introducing an algorithm which simultaneously numerically constructs the form and proves its transformation laws. We also explicitly construct the 7th order form and prove analytically that it has the proper modular transformations. It is conjectured the same method will apply for other orders.

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

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