Some determinants of path generating functions, II

TL;DR

This paper investigates Hankel determinants of matrices formed from path generating functions, revealing that the determinant of Motzkin prefix numbers remains 1 regardless of matrix size, with various corollaries.

Contribution

It provides new evaluations of Hankel determinants for path generating functions and establishes a surprising invariance for Motzkin prefix numbers.

Findings

Hankel determinants of path generating functions are explicitly evaluated.

The Hankel determinant of Motzkin prefix numbers is always 1.

Multiple corollaries follow from these determinant evaluations.

Abstract

We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.