Hankel continued fraction and its applications

Abstract

Abstract. The Hankel determinants of a given power series f can be evaluated by using the Jacobi continued fraction expansion of f . However the existence of the Jacobi continued fraction needs that all Hankel determinants of f are nonzero. We introduce Hankel continued fraction, whose existene and unicity are guaranteed without any condition for the power series f . The Hankel determinants can also be evaluated by using the Hankel continued fraction.It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for power series. If a power series f over a finite field satisfies a quadratic functional equation, then the Hankel continued fraction is ultimately periodic. As an application, we derive the Hankel determinants of several automatic sequences, in particular, the regular paperfolding sequence. Thus we provide an automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured by Coons-Vrbik.

Hankel continued fractions and Hankel determinants of the Euler numbers

“…The Hankel determinants can also be evaluated by using the Hankel continued fraction. Let us recall the basic definition and properties of the Hankel continued fractions [21]. Definition 2.1.…”

“…(ii) Let F (x) be a power series such that its H-fraction is given by (2.10) with δ = 2. Then, all non-vanishing Hankel determinants of F (x) are given by [21,47,5,23] for the proof of Theorem 2.2.…”

“…The present form appeared for the first time in the Ph.D. thesis by Emmanuel Roblet in 1994 [47, p. 44-56] 2 . It was independently rediscovered by Buslaev in 2010 [5], by Boltz-Tyaglov in 2012 [23], and by the author in 2016 [21]. The name of the continued fraction called by Roblet and Buslaev is P -fraction.…”

“…This implies that the corresponding Jacobi continued fraction does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction [21], instead of the J-fraction. Consequently, we derive an explicit formula for the Hankel determinants of the Euler numbers.…”

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

“…In [22,23], van der Poorten also studied continued fraction expansions for other infinite products. In 2016, Han [17] proved an analogue of Lagrange's theorem for Hankel continued fractions of quadratic power series on finite fields. Now we consider the continued fraction given an automatic sequence as the sequence of partial quotients.…”