Finding both, the continued fraction and the Laurent series expansion of golden ratio analogs in the field of formal power series

Abstract

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Knowing both of them is for instance important for the study of Kronecker type sequences in the theory of uniform distribution.Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions. We discover that both matrices in the LU decomposition of these Hankel matrices can be described by a simple recursive algorithm based on the continued fraction coefficients of the golden ratio analogs.The upper triangular factor U possesses several nice properties. First, its entries in the special case where the golden ratio analog is [0;X‾] can be given by using the Catalan's triangular numbers. Nicely, this relation together with our findings on the Hankel matrices are used to derive combinatorial identities involving Catalan's triangular numbers. As a side product we obtain a description of the distribution of the Catalan numbers modulo a prime number.Second, the upper triangular factor U is the columnwise composition of the Zeckendorf-type representations of the powers of X. This Zeckendorf representation for polynomials is introduced in the style of the Zeckendorf representation of the natural numbers and it is based on the Fibonacci polynomials instead of Fibonacci numbers.Third, in the case of positive characteristic, for each purely periodic golden ratio analog we detect a self-similar pattern in the upper triangular factor U of its Hankel matrix. We derive this pattern from a binomial type formula for two non-commutative matrices.Finally, we exemplary show, how this self-similar pattern in the upper triangular factor U can be used to describe the Laurent series expansion of the golden ratio analog [0;X‾].