Hankel determinants for convolution powers of Motzkin numbers

TL;DR

The work addresses the problem of evaluating Hankel determinants $H_n(F(x,r))$ for the convolution powers $F(x,r)=M(x)^r$ of Motzkin numbers, using generating-function techniques. The main approach leverages Sulanke-Xin's continued-fraction method and the quadratic transformation $\tau$ to relate $H(F)$ to $H(\tau(F))$, enabling shifted periodic continued fractions and tractable recursions. For $r=3$ and $r=4$, the paper derives explicit recursions and initial data via these transformations, and extends the pattern to conjectures for general $r$, verified computationally up to $r\le 27$. The contributions include closed-form-like determinant relations in several residue classes of $r$, a systematic computational pipeline, and a set of conjectures guiding future analytic proofs and extensions to Motzkin-related combinatorics.

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.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Conjecture 3
• Theorem 4
• Theorem 5
• Theorem 6
• Conjecture 7
• Conjecture 8
• Theorem 9
• Proposition 10