Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces

For all integers $k>0$, we prove that the hypergeometric function \[ \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of pencil curves genus $3k+1$. We $\widehat {I}_k$ generating Gromov–Witten invariants family anticanonical del Pezzo hypersurfaces $X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1)$. Thus, Landau–Ginzburg mirror family. The surfaces $X$ were first constructed by Johnson and Kollár. feature these makes our construction especially interesting $|-K_X|=|\mathcal {O}_X (1)|=\varnothing$. This means there no way to form Calabi–Yau pair $(X,D)$ out hence known for other than one given here. also discuss connection between work Beukers, Cohen Mellit on functions.