Tree-indexed sums of Catalan numbers

TL;DR

This paper introduces a method to evaluate infinite sums of products of Catalan numbers indexed by trees, expressing them as polynomials in 1/π with rational coefficients, and relates them to elliptic integrals.

Contribution

It provides an effective algorithm for computing these sums and introduces parametric liftings linking them to elliptic integrals, advancing the understanding of tree-indexed Catalan sums.

Findings

Sums are polynomials in 1/π with rational coefficients.

Introduces parametric liftings related to elliptic integrals.

Degrees of polynomials are at most half the number of tree vertices.

Abstract

We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/\pi$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.