  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 $A_{1}^{(1)}$ in terms of
Dedekind eta functions. We produce new relations between string functions by
writing them as double-sums and then using certain symmetry relations. We
evaluate the series using special double-sum formulas that express Hecke-type
double-sums 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.
