A note on the extended dispersionless Toda hierarchy

We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological $\mathbb CP^1$ model. Using the dispersionless Hirota equations, we compute the two-point functions and express the result in terms of Catalan numbers.