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

We remember the golden ratio $\varphi:=\frac{\sqrt{5}+1}{2}$ as a root of the polynomial $p[Y]\in\mathbb{Q}[Y]$

and its fractional part $\{\varphi\}=\frac{\sqrt{5}-1}{2}=1/\varphi$ as a root of the polynomial $\overline{p}[Y]\in\mathbb{Q}[Y]$

It is well-known that both $\varphi$ and $1/\varphi$ are quadratic irrationals in $\mathbb{R}$ which is the completion field of $\mathbb{Q}$ with respect to the absolute value $|\,\cdot\,|$.

The golden ratio $\varphi$ and its inverse $1/\varphi$ are in a certain sense optimal real numbers, namely their simple continued fraction coefficients have lowest possible absolute values $1$, as $\varphi=[1;\overline{1}]$ and $1/\varphi=[0;\overline{1}]$, and the latter is contained in the interval $[0,1)$.
Note that the $n$th convergent to $\varphi$ is $\frac{F_{n+1}}{F_{n}}$ and to $1/\varphi$ is $\frac{F_{n-1}}{F_{n}}$, where $F_{n}$ denotes the $n$th Fibonacci number. The sequence of Fibonacci numbers $(F_{n})_{n\geq 0}$ is recursively defined via $F_{n}=F_{n-1}+F_{n-2}$ where we choose the starting values $F_{0}=1$ and $F_{-1}=0$.

Now we use – in analogy to the number field $\mathbb{Q}$ – the rational function field $\mathbb{K}=\boldsymbol{k}(X)$ where $\boldsymbol{k}$ is an arbitrary field, as for example $\mathbb{Q}$ or a finite field $\mathbb{F}_{q}$ with $q$ elements. And the pendant to the absolute value is the non-archimedean absolute value $|\,\cdot\,|_{\infty}$, that is defined using the degree valuation $\nu_{\infty}$ as follows.

For every $f\in\boldsymbol{k}(X)$ we can find $p(X),\,q(X)\in\boldsymbol{k}[X]$ satisfying $q(X)\neq 0$ such that $f=\frac{p(X)}{q(X)}$. We write $q(X)=\sum_{i=0}^{r}a_{i}X^{i}$ with $a_{i}\in\boldsymbol{k}$ and $a_{r}\neq 0$. Then the degree $\deg(q(X))$ of $q(X)$ is well-known to be $r$. We define $\deg(0):=-\infty$.
The degree valuation $\nu_{\infty}$ of the rational function $f$ then is defined as

and the corresponding non-archimedean absolute value

where $e$ is here the Euler number.

Now the completion field of $\boldsymbol{k}(X)$ with respect to the absolute value $|\,\cdot\,|_{\infty}$ is the field of formal Laurent series (also called formal power series), denoted by $\boldsymbol{k}((X^{-1}))$. Each element $L\neq 0$ in $\boldsymbol{k}((X^{-1}))$ can be written as

here $c_{i}\in\boldsymbol{k}$ for all $i\in\mathbb{Z}$ and $u\in\mathbb{Z}$ such that $c_{u}\neq 0$ and $c_{i}=0$ for all $i<u$.

The degree evaluation $\nu_{\infty}(L)$ of $L$ is $-u$ and the absolute value $|L|_{\infty}$ of $L$ is $e^{-u}$.

In the following we list analogies between $\mathbb{R}$ and $\boldsymbol{k}((X^{-1}))$ that are of use in this paper.

1. 1.

   Let $b\in\mathbb{Z}$ satisfying $b>1$. Then every real number $\alpha$ has a unique $b$-adic expansion of the form

   $$[\alpha]+\sum_{i=1}^{\infty}c_{i}b^{-i}$$

   with $c_{i}\in\{0,1,\ldots,b-1\}$ and infinitely many $c_{i}\neq b-1$, and $[\alpha]$ is the integer part of $\alpha$ satisfying $\alpha-[\alpha]\in[0,1)$. The latter magnitude is abbreviated to $\{\alpha\}$ and called the fractional part of $\alpha$.
   

   The element $X\in\boldsymbol{k}[X]$ satisfies $|X|_{\infty}=e^{1}>1$. Every nonzero element $L$ in $\boldsymbol{k}((X^{-1}))$, can be uniquely written as

   $$[L]+\sum_{i=1}^{\infty}c_{i}X^{-i}$$

   with coefficients $c_{i}\in\boldsymbol{k}$, and $[L]=\sum_{i=0}^{-u}{c_{-i}}X^{i}$ is the polynomial part of $L$ contained in $\boldsymbol{k}[X]$. Note that the absolute value of $|L-[L]|_{\infty}$ is smaller than $1$. The element $(L-[L])$ is usually abbreviated to $\{L\}$ and called the fractional part of $L$.

   The set of all possible fractional parts, that is

   $$\{L\in\boldsymbol{k}((X^{-1})):|L|_{\infty}<1\},$$

   will be abbreviated to $\overline{\boldsymbol{k}((X^{-1}))}$ in the following.

   Note that this set can be alternatively given as

   $$\left\{\sum_{i=1}^{\infty}c_{i}X^{-i}:c_{i}\in\boldsymbol{k}\right\}.$$

   The degree evaluation $\nu_{\infty}$ of $\sum_{i=1}^{\infty}c_{i}X^{-i}\neq 0$ satisfies

   $$\nu_{\infty}\left(\sum_{i=1}^{\infty}c_{i}X^{-i}\right)=-\min\left(\{i\in\mathbb{N}:c_{i}\neq 0\}\right).$$

2. 2.

   Every real number $\alpha\in[0,1)$ can be expressed in a continued fraction $[0;a_{1},a_{2},a_{3},\ldots]$ satisfying $a_{i}\in\mathbb{N}$ for every $i\in\mathbb{N}$.
   

   In analogy, every series $L\in\overline{\boldsymbol{k}((X^{-1}))}$ can be expressed in a continued fraction

   $$[0;A_{1}(X),A_{2}(X),A_{3}(X),\ldots]=\cfrac{1}{A_{1}(X)+\cfrac{1}{A_{2}(X)+\cfrac{1}{A_{3}(X)+\ddots}}}$$

   with $A_{i}(X)\in\boldsymbol{k}[X]$ satisfying $\deg(A_{i}(X))\geq 1$ for every $i\in\mathbb{N}$.

3. 3.

   The $n$th convergent $\alpha_{n}$ to a non-rational $\alpha\in[0,1)$ is the finite continued fraction $[0;a_{1},a_{2},\ldots,a_{n}]$ which can be written as $p_{n}/q_{n}$ with $p_{n},q_{n}\in\mathbb{N}_{0}$. Both recursively computed via $q_{n}=a_{n}q_{n-1}+q_{n-2}$ and $p_{n}=a_{n}p_{n-1}+p_{n-2}$ for every $n\in\mathbb{N}$ with starting values $q_{-1}=0,\,p_{-1}=1,\,q_{0}=1$, and $p_{0}=0$.
   Analogously, the $n$th convergent $L_{n}\in\boldsymbol{k}(X)$ to a non-rational $L\in\overline{\boldsymbol{k}((X^{-1}))}$ is the finite continued fraction $[0;A_{1}(X),A_{2}(X),\ldots,A_{n}(X)]$ which can be written as $P_{n}(X)/Q_{n}(X)$ with $P_{n}(X),Q_{n}(X)\in\boldsymbol{k}[X]$. Both recursively computed via $Q_{n}(X)=A_{n}(X)Q_{n-1}(X)+Q_{n-2}(X)$ and $P_{n}(X)=A_{n}(X)P_{n-1}(X)+P_{n-2}(X)$ for every $n\in\mathbb{N}$ with initial values $Q_{-1}(X)=0,\,P_{-1}(X)=1,\,Q_{0}(X)=1$, and $P_{0}(X)=0$.

4. 4.

   Now an interesting fact is that the denominators $q_{n}$ or $Q_{n}(X)$ are best approximating in the sense that if

   $$|\alpha-p/q|<1/|q|^{2}$$

   then $p/q$ is a convergent to $\alpha$ and if

   $$\nu_{\infty}(L-P(X)/Q(X))<2\nu_{\infty}\left(\frac{1}{Q(X)}\right)=-2\deg(Q(X))$$

   then $P(X)/Q(X)$ is a convergent to $L$.

   Moreover, we note that $\deg(Q_{n})=\sum_{i=1}^{n}\deg(A_{i})=:d_{n}$, $\nu_{\infty}(L-P_{n}(X)/Q_{n}(X))=-d_{n}-d_{n+1}$ and

   $$\nu_{\infty}(L-v(X)/w(X))\geq\nu_{\infty}(L-P_{n}(X)/Q_{n}(X))$$

   for all $v(X),w(X)\in\boldsymbol{k}[X]$ satisfying $0\leq\deg(w(X))<d_{n+1}$.

5. 5.

   Finally, we point out two further analogies between $\mathbb{R}$ and the set of formal power series $\boldsymbol{k}((X^{-1}))$ in the case where $\boldsymbol{k}$ is a finite field. We have the following two classifications of $\alpha\in\mathbb{R}$:

   • -

     $\alpha$ is a rational number if and only if for every integer $b>1$, the $b$-adic expansion of $\alpha$ is eventually periodic or finite.

   • -

     $\alpha$ is a quadratic irrational number if and only if the continued fraction expansion of $\alpha$ is infinite and ultimately periodic.

   In the case where the field $\boldsymbol{k}$ is finite, we have for $L\in\boldsymbol{k}((X^{-1}))$ that the condition $L\in\boldsymbol{k}(X)$ is equivalent to that the series expansion $\sum_{i=u}^{\infty}c_{i}X^{-i}$ is eventually periodic or finite. (See [7, Theorem 1.1])

   Moreover, in case of a finite field $\boldsymbol{k}$ we have for $L\in\boldsymbol{k}((X^{-1}))\setminus\boldsymbol{k}(X)$, that the continued fraction expansion of $L$ is ultimately periodic if and only if it is quadratic over $\boldsymbol{k}(X)$ (cf. e.g. [7, Theorem 3.1]).

   For results and discussions in the case where $\boldsymbol{k}$ is not finite, confer e.g. [10].

For more information on continued fractions in power series fields we refer to [7] and [10].

In this paper we introduce analogs of the reciprocal of the golden ratio $1/\varphi$ which possesses the “optimal” continued fraction expansion

since all coefficient have lowest possible value $1$. As there are more choices of partial quotients having lowest possible degree $1$, we define a set of golden ratio analogs in $\overline{\boldsymbol{k}((X^{-1}))}$.

We are particularly interested in the formal power series representation of specific examples of golden ratio analogs, as e.g. $[0;\overline{X}]$ over special fields $\boldsymbol{k}$. We would like to point out that for real numbers in $[0,1)$ it is a non-trivial problem to determine both, the explicit $b$-adic representation with a fixed base $b$, say $2$ or $10$, and its continued fraction coefficients. So far there are only a few examples, where we have information on both representations (see e.g. Shallit [11, 12]). We mention here the number investigated in [11].

The question for the formal series expansion of golden ratio analogs is for instance relevant for the investigation of so-called Kronecker type sequences over $\mathbb{F}_{q}((X^{-1}))$ where $\mathbb{F}_{q}$ is the finite field with $q$ elements. Kronecker type sequences, that are introduced as an analogon to the ordinary Kronecker sequence $(\{n\alpha\})_{n\geq 0}$ with $\alpha\in\mathbb{R}$, are actively investigated in the theory of uniform distribution (cf e.g.[5]). For the sake of completeness we give the definition of Kronecker type sequences.

In the theory of uniform distribution the so-called $L_{2}$-discrepancy of Kronecker type sequences is of particular interest as the $L_{2}$-discrepancy of ordinary Kronecker sequences is well investigated (cf. for example [1]). The main method for investigating the distribution of Kronecker type sequences applies Walsh functions whereas for the ordinary Kronecker sequence $(\{\alpha n\})_{n\geq 0}$ trigonometric functions are used. For the Walsh function approach detailed information on the formal series coefficients is helpful. It is known that one-dimensional Kronecker type sequences $([\{n(X)L\}]_{q})_{n\geq 0}$ are in a certain sense optimal if all continued fraction coefficients of $L$ have degree $1$ (see e.g. [9, Theorem 4.48]). Therefore the series expansions of golden ratio analogs in $\overline{\boldsymbol{k}((X^{-1}))}$, where $\boldsymbol{k}$ is a finite field, deserve particular attention.
As already noted in the abstract our approach studies the Hankel matrices that are the generating matrices of the Kronecker type sequences and which are defined using the coefficients of the Laurent series expansions.

We call a matrix $\mathcal{C}\in\boldsymbol{k}^{\mathbb{N}\times\mathbb{N}}$ regular if for every $k\in\mathbb{N}$ its upper left $k\times k$ submatrix is regular.

In the following lemma we write down a basic property of Hankel matrices over finite fields, that is a consequence of [6, Corollary 1] and points out the relevance of the golden ratio analogs.

In the rest of the paper we aim for explicit formulas for the coefficients in the series expansions of golden ratio analogs, in particular for the specific case where $\boldsymbol{k}$ is a finite field $\mathbb{F}_{q}$.

We start with the most basic example $[0;\overline{X}]$ over $\mathbb{F}_{2}$.

Note that over a more general field with characteristic greater than $2$ or zero, squaring a series is not that trivial. It is ad hoc not clear how to find the expansion of $[0;\overline{X}]$ over e.g. $\mathbb{F}_{p}$ where $p$ is an odd prime number. The method developed in this paper is based on studying the associated Hankel matrices. As side products we discover plenty of interesting and astonishing results. Just to announce two of them. We give the unique $\mathcal{L}_{\phi}\mathcal{U}_{\phi}$ decomposition of the Hankel matrix $\mathcal{M}_{\phi}$ for any golden ratio analog $\phi$ over $\boldsymbol{k}$, where the columns of $\mathcal{U}_{\phi}$ and rows of $\mathcal{L}_{\phi}$ follow easy to define recursions. We identify for specific examples of golden ratio analogs $\phi$ nice patterns in $\mathcal{U}_{\phi}$, that will give us a hint on the Laurent series expansion of $\phi$.

Throughout this section $\boldsymbol{k}$ and $\phi\in\Phi\subseteq\overline{\boldsymbol{k}((X^{-1}))}$ will be fixed. Thus the continued fraction $[0;A_{1}(X),A_{2}(X),A_{3}(X),\ldots]$ of $\phi$ satisfies $\deg(A_{i}(X))=1$ for all $i\in\mathbb{N}$.

We define the sequence of Fibonacci polynomials $\mathcal{F}_{\phi}$ associated with $\phi$ given by the denominators of the convergents to $\phi$.

We observe that all Fibonacci polynomials associated with $\phi\in\Phi$ satisfy $\deg(F_{n}(X))=n$ for all $n\in\mathbb{N}_{0}$. Hence we are able to define a so-called “Zeckendorf representation” in $\boldsymbol{k}[X]$. We note that the sequence of Fibonacci numbers $(F_{n})_{n\geq 0}$ in $\mathbb{N}_{0}$ can be used to represent any non-negative integers $m$ in terms of different Fibonacci numbers, i.e. $m=\sum_{i=1}^{\infty}\delta_{i}F_{i}$ with $\delta_{i}\in\{0,1\}$ and only finitely many $\delta_{i}=1$. For the Zeckendorf representation we require, if $\delta_{i}=1$ then $\delta_{i+1}=0$. The latter guaranties the uniqueness of this representation.

Let $\mathcal{F}_{\phi}:=(F_{n}(X))_{n\geq 0}$ be the sequence of Fibonacci polynomials associated with $\phi\in\Phi$ and $(f_{n})_{n\geq 0}$ is the sequence of the leading coefficients in $\boldsymbol{k}$, i.e., $f_{n}={\rm lc}(F_{n}(X))$.

Let $P(X)\in\boldsymbol{k}[X]$ with $\deg(P(X))=r\in\mathbb{N}_{0}$. Usually, $P(X)$ is expressed in terms of powers of $X$, i.e. $P(X)=\sum_{i=0}^{r}p_{i}X^{i}$ with $p_{i}\in\boldsymbol{k}$ for $i=0,\ldots,r$ and $p_{r}\neq 0$. By the following greedy algorithm one can represent $P(X)$ in terms of the Fibonacci polynomials, i.e.,

with $z_{i}\in\boldsymbol{k}$ and $z_{r}\neq 0$.

The following lemma points out an important advantage of the Zeckendorf representation of a polynomial $n(X)$ in $\boldsymbol{k}[X]$.

Next we define matrices associated to a golden ratio analog $\phi$ based on its continued fraction coefficients.

Note that in the special case where $\boldsymbol{k}=\mathbb{F}_{2}$, $\mathcal{R}_{\phi}^{T}=\mathcal{R}_{\phi}$ and as a consequence $\mathcal{L}_{\phi}^{T}=\mathcal{U}_{\phi}$. In the following theorem we exhibit specific properties of $\mathcal{L}_{\phi}$ and $\mathcal{U}_{\phi}$.