On string functions and double-sum formulas

String functions are important building blocks of characters integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string for Lie algebras type $A_{1}^{(1)}$ in terms Dedekind eta functions. We produce new relations between by writing them as double-sums then using certain symmetry relations. evaluate the series special double-sum formulas that express Hecke-type Appell--Lerch theta functions, where we point out Ramanujan's classical mock