Catalan Moments

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.