Eric T. Mortenson’s research while affiliated with Saint Petersburg State University and other places

Kac and Wakimoto introduced the admissible highest weight representations in order to classify all modular invariant representations of the Kac--Moody algebras. Ahn, Chung, and Tye studied the string functions of the admissible highest weight representations of $A_1^{(1)}$ and realized them as the characters of the generalized parafermionic theories. These string functions are allowed to have certain positive and negative rational levels, for integer levels they reduce to the well-studied string functions of the integrable highest weight representations of Kac and Peterson. In this paper we demonstrate that the mock modular part of 1/2-level string functions can be evaluated in terms of Ramanujan's mock theta functions by using recent Hecke-type double-sum formulas of positive discriminant (Mortenson and Zwegers, 2023). We also present elegant mock theta conjecture-like identities and mixed mock modular properties for certain examples of 1/2-level string functions. In addition, we show that the negative level string functions can be evaluated in terms of false theta functions by using recent Hecke-type double-sum formulas of negative discriminant (Mortenson, 2024).

Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews–Crandall-type identity and use it to count the number of integer solutions to x2+2y2+2z2=n$x^2+2y^2+2z^2=n$.

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.

Using Appell function properties we give short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$ after work of Chan and Mao; Mao; and Brietzke, da Silva, and Sellars. We also present a generalization of the identities in the spirit of celebrated results of Bringmann, Ono, and Rhoades on Dyson's ranks and Maass forms.

Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews--Crandall-type identity and use it to count the number of integer solutions to $x^2+2y^2+2z^2=n$.

We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell–Lerch functions—the building blocks of Ramanujan’s mock theta functions—can be used to give general string function formulas for the affine Lie algebra A1(1)$A_{1}^{(1)}$ for levels N=1,2,3,4N=1,2,3,4.

Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. 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 over eight sixth-order mock theta functions in the lost notebook, but among their many identities there is only one relationship like those of the tenth-orders. Recently, three new identities for the sixth-order mock theta functions that are in the spirit of the six tenth-order identities were discovered. Here we present several families of tenth-order like identities for Appell functions, which are the building blocks of Ramanujan's mock theta functions.

String functions are important building blocks of characters of integrable highest modules over affine Kac–Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type A1(1)$A_{1}^{(1)}$ in terms of Dedekind eta functions. We obtain new symmetries for string functions by exploiting their natural setting of Hecke-type double-sums, where special double-sums are expressed in terms of Appell–Lerch functions and theta functions, where we point out that Appell–Lerch functions are the building blocks of Ramanujan’s classical mock theta functions. We then demonstrate the utility of the new symmetries by giving new proofs of classical string function identities.