An algebraic interpretation of the super Catalan numbers (Q6564342)

The author develops a polynomial integration theory over circles in Euclidean geometry for general fields \(F\) of characteristic zero, without relying on limiting processes (such a theory was previously established in positive odd characteristics). This algebraic integral over a circle \(C\) is defined as an \(F\)-linear functional \(\phi :F[\alpha_1, \alpha_2]\to F \) on the polynomial ring \(F[\alpha_1, \alpha_2]\) such that \(\phi (1)=1\) , \( \phi (\pi )=0\) for any polynomial function \(\pi\) vanishing on the circle \(C, \) while also possessing the property of rotational invariance. The main result of the paper states that for any \(k, l\in \mathbb N,\) the linear functional \(\psi\) defined by the rule: \(\psi (\alpha_1^k\alpha_2^l)=\Omega (m, n)\) if \(k=2m\) and \(l=2n\) and \( \psi (\alpha_1^k\alpha_2^l)=0 \) otherwise is the unique circular integral functional with respect to \(C\) (here \( \Omega (m, n)=S(m, n)/4^{m+n},\) where \(S(m, n)=(2m)!(2n)!/(m!n!(m+n)!) \) is the super Catalan number). This result gives an algebraic interpretation of super Catalan numbers (no combinatorial interpretation of \(S(m, n)\) is known for general \(m\) and \(n\), in contrast to over \(200\) interpretations of the Catalan numbers).