Motzkin Numbers: an Operational Point of View

TL;DR

This paper explores Motzkin numbers through an operational perspective, deriving new identities and uncovering links with special functions using umbral methods, thus opening new avenues for research and generalizations.

Contribution

It introduces an operational approach to Motzkin numbers, deriving new identities and establishing connections with special functions and umbral calculus.

Findings

Derived new identities for Motzkin numbers

Linked Motzkin numbers with special functions

Identified new Motzkin-related forms

Abstract

The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory of special functions and with the relevant treatment in terms of operational means. The use of umbral methods opens new directions for further developments and generalizations, which leads, e.g., to the identification of new Motzkin associated forms.