On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths

TL;DR

This paper introduces dual polynomial sequences related to Fibonacci and Catalan-Fibonacci polynomials using coefficient array inversion, revealing connections to Motzkin path enumeration and Riordan arrays.

Contribution

It defines new dual polynomials via array inversion and explores their properties, linking them to Motzkin path statistics and Riordan arrays.

Findings

Dual polynomials are characterized and their properties analyzed.

Connections established between these polynomials and Motzkin path statistics.

Many involved arrays are identified as Riordan arrays.

Abstract

We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.