Eichler-Selberg relations for the third-order mock theta functions

Ramanujan’s third-order mock theta function $f(q)$ admits the well-known Appell-Lerch series representation: $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{(-q;q{)}_n^2}=\frac{2}{(q;q{)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(-1{)}^n{q}^{\frac{3}{2}{n}^2+\frac{1}{2}n}}{1+q^n}\mathrm{.}$$ In this paper, we establish a natural generalization of this classical identity by utilizing the theory of harmonic Maass forms. Furthermore, we prove analogous Eichler-Selberg type relations for another third-order mock theta function $\omega (q)$. The method presented in this paper can be extended to study other classes of mock theta functions.