For all integers k > 0, we prove that the hypergeometric function
Ibk(α) = X∞ j=0 (8k + 4)j [divided by] !j! (2j)! (2k + 1)j!2 (4k + 1)j! αj
is a period of a pencil of curves of genus 3k+1. We prove that the function Ibk is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X8k+4 ⊂ P(2, 2k + 1, 2k + 1, 4k + 1). Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X were first constructed by Johnson and Koll´ar. The feature of these surfaces that makes our mirror construction especially interesting is that | − KX| = |OX(1)| = ∅. This means that there is no way to form a Calabi–Yau pair (X, D) out of X and hence there is no known mirror construction for X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.