Hankel determinants for convolution powers of Motzkin numbers

TL;DR

This paper computes Hankel determinants for convolution powers of Motzkin numbers using shifted periodic continued fractions and proposes conjectures on their polynomial characterization.

Contribution

It introduces a novel application of continued fraction methods to evaluate Hankel determinants of convolution powers of Motzkin numbers and presents new conjectures.

Findings

Hankel determinants evaluated for convolution powers up to r=27

Identification of shifted periodic continued fractions in the analysis

Conjectures on polynomial characterization of determinants

Abstract

We evaluate the Hankel determinants of the convolution powers of Motzkin numbers for $r\leq 27$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin's continued fraction method. We also conjecture some polynomial characterization of these determinants.