Hankel determinants and shifted periodic continued fractions

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), ( −1, −1)} for integer parameters m, q, . Again shifted periodic continued fractions appear.Mathematic subject classification: Primary 15A15; Secondary 05A15, 11B83.

Hankel determinants for convolution powers of Catalan numbers

“…Sulanke-Xin's quadratic transformation τ . This subsection is copied from [29]. We include it here for reader's convenience.…”

“…was found in [29] to appear in Hankel determinants of many path counting numbers. Here p is an additional parameter.…”

Dual Numbers, Weighted Quivers, and Extended Somos and Gale-Robinson Sequences

“…Sequences with Hankel determinants consisting of 0, 1 and ´1 were considered in [8,5,6,41]; according to [8] the question of characterization of such sequences was asked by Michael Somos. We contribute to this study.…”

“…In Section 4.2 we will prove the following property of the q-deformed golden ratio (1.4). [41,5,7,13] and [21].…”