Catalan Moments

TL;DR

This paper explores relations among certain operators and their actions on sequences, revealing connections to Catalan and Motzkin numbers, and introduces new identities involving these classical combinatorial sequences.

Contribution

It demonstrates conjugation relations among operators in a algebraic group, analyzes their effects on recurrence sequences, and links these to orthogonal polynomials and Catalan numbers, providing new identities.

Findings

Operators are conjugated in the algebraic group 0psilon(R).

Moments of orthogonal polynomials relate to generalized Motzkin and Catalan numbers.

New identities on Catalan numbers derived from orthogonality relations.

Abstract

This paper is essentially devoted to the study of some interesting relations among the well known operators $I^{(x)}$ (the interpolated Invert), $L^{(x)}$ (the interpolated Binomial) and Revert (that we call $\eta$). We prove that $I^{(x)}$ and $L^{(x)}$ are conjugated in the group $\Upsilon(R)$. Here $R$ is a commutative unitary ring. In the same group we see that $\eta$ transforms $I^{(x)}$ in $L^{(-x)}$ by conjugation. These facts are proved as corollaries of much more general results. Then we carefully analyze the action of these operators on the set $\mc{R}$ of second order linear recurrent sequences. While $I^{(x)}$ and $L^{(x)}$ transform $\mc{R}$ in itself, $\eta$ sends $\mc{R}$ in the set of moment sequences $\mu_n(h,k)$ of particular families of orthogonal polynomials, whose weight functions are explicitly computed. The moments come out to be generalized Motzkin numbers (if $R=\zz$, the Motzkin numbers are $\mu_n(-1,1)$). We give several interesting expressions of $\mu_n(h,k)$ in closed forms, and one recurrence relation. There is a fundamental sequence of moments, that generates all the other ones, $\mu_n(0,k)$. These moments are strongly related with Catalan numbers. This fact allows us to find, in the final part, a new identity on Catalan numbers by using orthogonality relations.