An Algebraic Interpretation of the Super Catalan Numbers

We extend the notion of polynomial integration over an arbitrary circle $C$ in the Euclidean geometry over general fields $\mathbb F$ of characteristic zero as a normalized $\mathbb F$-linear functional on $\mathbb{F}\left[\alpha_1, \alpha_2\right]$ that takes polynomials that evaluate to zero on $C$ to zero and is $\mathrm{SO}(2,\mathbb{F})$-invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!}$$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha_1^{2m}\alpha_2^{2n}$.