Catalan Moments
Stefano Barbero, Umberto Cerruti

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 (the interpolated Invert), (the interpolated Binomial) and Revert (that we call ). We prove that and are conjugated in the group . Here is a commutative unitary ring. In the same group we see that transforms in 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 of second order linear recurrent sequences. While and transform in itself, sends in the set of moment sequences of particular families of orthogonal polynomials, whose weight functions are explicitly computed. The moments come out to be generalized Motzkin numbers…
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Taxonomy
TopicsSpanish Literature and Culture Studies · Spanish History and Politics · Medieval European Literature and History
