On Ramanujan's lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions

Abstract

Ramanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth-order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. We also find eight sixth-order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth-orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth-order mock theta functions that are in the spirit of the six tenth-order identities. We also include an additional nineteen tenth-order like identities for various combinations of second-, sixth-, and eighth-order mock theta functions.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Proposition 3.1
• Corollary 3.2