Determining the explicit forms and modularity for string functions and branching coefficients for Kac–Moody algebras after Kac, Peterson, and Wakimoto is an important problem. For positive admissible-level string functions for the affine Kac–Moody algebra \(A_{1}^{(1)}\) , very little is known. Here we apply the notion of quasi-periodicity to a generalized Euler identity of Schilling and Warnaar for the affine Kac–Moody algebra \(A_{1}^{(1)}\) . For integral-level string functions, the classical periodicity reduces the infinite sum of string functions in the generalized Euler identity to a finite sum of string functions with theta function coefficients. For admissible level, we similarly reduce to an analogous finite sum of string functions, but we also gain an additional finite sum of the form \(\begin{aligned} \sum _{i}\Phi _{i}(q)\Psi _{i}(q), \end{aligned}\) where the \(\Phi _i(q)\) ’s are modular and depend only on the spin, and the \(\Psi _{i}(q)\) ’s are (mixed) mock modular Hecke-type double-sums and depend only on the quantum number. For levels 1/2, 1/3, and 2/3, we shall also see that the \(\Psi _{i}(q)\) ’s give us families of mock theta conjecture-like identities for symmetric Hecke-type double-sums. Our work here focuses on evaluating the \(\Psi _{i}(q)\) ’s, and our expressions utilize Ramanujan’s second-order mock theta function \(\mu _2(q)\) and third-order mock theta functions \(f_{3}(q)\) , \(\omega _3(q)\) , \(\psi _{3}(q)\) , and \(\chi _3(q)\) .
