Finite Interpretation of the Hyper-Catalan Series Zero and its Powers

Abstract: In 2025, Wildberger and Rubine showed the formal series zero of the univariate geometric polynomial is $\mathbf{S}$, the generating series for the hyper-Catalan numbers $\mathbf{C}_m$, which count the number of roofed subdivided polygons (subdigons) of type $\mathbf{m}$. We show that we can interpret this result as a finite identity at each level, where a level is a truncation of $\Sb$ to a given maximum number of vertices, edges, or faces (bounded by degree) of the associated subdigon types. We then explore powers $\mathbf{S}^r$, recounting Raney's and our own combinatorial derivations of its coefficients.