BCOV cusp forms of lattice polarized K3 surfaces

We introduce the BCOV formula for the lattice polarized K3 surfaces. We find that it yields cusp forms expressed by certain eta products for many families of rank 19 lattice polarized K3 surfaces over $\mathbb{P}^{1}$. Moreover, for Clingher-Doran's family of $U\oplus E_{8}(-1)\oplus E_{7}(-1)$-polarized K3 surfaces, we obtain the Igusa cusp forms $\chi_{10}$ and $\chi_{12}$ from the formula. Inspired by the arithmetic properties of mirror maps studied by Lian-Yau, we also derive the K3 differential operators for all the genus zero groups of type $\Gamma_{0}(n)_{+}$.