Harmonic shifted symmetric polynomials and the Bloch-Okounkov theorem

The Bloch-Okounkov theorem yields a $\eufm{sl}_2$-equivariant map defined by sums over partitions from shifted symmetric polynomials to quasimodular forms. Classical modular forms are precisely given by the kernel of one of the operators of this $\eufm{sl}_2$-triple on quasimodular forms. The inverse image of modular forms, which is the kernel of another operator $\Delta$, is called the space of harmonic shifted symmetric polynomials. We find an explicit basis for this space using an analogue of the Kelvin transform from the theory of harmonic functions.