The H and K Family of Mock Theta Functions

Abstract In his last letter to Hardy, Ramanujan defined 17 functions F ( q ), | q | < 1, which he calledmock θ -functions. He observed that as q radially approaches any root of unity ζ at which F ( q ) has an exponential singularity, there is a θ -function T ζ ( q ) with F ( q ) − T ζ ( q ) = O (1). Since then, other functions have been found that possess this property. These functions are related to a function H ( x , q ), where x is usually q r or e 2π ir for some rational number r . For this reason we refer to H as a “universal” mock θ -function. Modular transformations of H give rise to the functions K , K 1 , K 2. The functions K and K 1 appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mock θ -functions of even order and H are listed.