Stringy $E$-functions of canonical toric Fano threefolds and their applications

Let $\mathrm{\Delta }$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_{\mathrm{\Delta }}$ associated with the polytope $\mathrm{\Delta }$. Using the stringy Libgober-Wood identity and our formula, we generalize the well-known combinatorial identity $\sum_{\theta \preceq \Delta \atop \dim (\theta) =1} v(\theta) \cdot v(\theta^*) = 24$holdinginthecaseof$3$-dimensional reflexive polytopes $\mathrm{\Delta }$.