Mock theta functions and Appell-Lerch sums.

Area of Science:

• Number Theory
• Special Functions
• Mock Theta Functions

Background:

• Mock theta functions are a class of q-series with complex properties.
• Appell-Lerch sums provide a framework for analyzing these functions.
• Previous work by Mortenson explored bilateral series for the universal mock theta function.

Purpose of the Study:

• To investigate bilateral series associated with the universal mock theta function.
• To express these series using Appell-Lerch sums.
• To uncover new relationships and congruences involving mock theta functions.

Main Methods:

• Utilizing the theory of Appell-Lerch sums.
• Developing and analyzing specific bilateral series representations.
• Applying techniques from number theory to establish congruence relations.

Main Results:

• Bilateral series for odd order mock theta functions are presented in terms of Appell-Lerch sums.
• A significant congruence relationship for the third order mock theta function is established.
• Internal relationships between two-group bilateral series of fifth order mock theta functions are derived.

Conclusions:

• The study extends the understanding of bilateral series for mock theta functions.
• New connections and properties of mock theta functions are revealed through Appell-Lerch sums.
• The findings contribute to the ongoing research in the field of special functions and number theory.