Hankel continued fractions and Hankel determinants for q-deformed metallic numbers · arXivDesk
2502.05993Feb 9, 202551 pages, 5 figures. Version 2 improvements: new theorem on the periodicity modulo a prime of the Hankel determinants, bibliography updates and minor corrections
Hankel continued fractions and Hankel determinants for q-deformed metallic numbers
Fix n a positive integer. Take the n-th metallic number φn=2n+n2+4
Nearby in the stack
(e.g.
φ1
is the golden number) and let
Φn(q)
be its
q
-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around
q=0
, with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first
n+2
sequences of shifted Hankel determinants of
Φn
and show that they satisfy the following properties: 1) They are periodic and consist of
−1,0,1
only. 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. 3) They are all completely determined by the first sequence. This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between
q
-deformations of real numbers and sequences of Catalan or Motzkin numbers.