  Motivated by the recent work of Kachru-Vafa in string theory, we study in
Part A of this paper, certain identities involving modular forms,
hypergeometric series, and more generally series solutions to Fuchsian
equations. The identity which arises in string theory is the simpliest of its
kind. There are nontrivial generalizations of the identity which appear new. We
give many such examples -- all of which arise in mirror symmetry for algebraic
K3 surfaces.
  In Part B, we study the integrality property of certain $q$-series, known as
mirror maps, which arise in mirror symmetry.
