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

On Hecke-type double-sums and general string functions for the affine Lie algebra $A_{1}^{(1)}$