  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 $\Gamma_{0}(n)_{+}$.
