Hankel determinants and shifted periodic continued fractions

TL;DR

This paper uses continued fraction methods to prove Hankel determinant conjectures and discovers new determinants related to lattice paths, highlighting the role of shifted periodic continued fractions in these evaluations.

Contribution

It provides short proofs of existing Hankel determinant conjectures and introduces new determinants linked to specific lattice paths, utilizing shifted periodic continued fractions.

Findings

Proofs of Cigler's Hankel determinant conjectures

Identification of shifted periodic continued fractions in computations

New Hankel determinants related to lattice paths with specific step sets

Abstract

Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set $\{(1,1),(q,0), (\ell-1,-1)\}$ for integer parameters $m,q,\ell$. Again shifted periodic continued fractions appear.