# On symmetric Tetranacci polynomials in mathematics and physics

Nico G. Leumer  1Donostia International Physics Center (DIPC), Paseo Manuel de
Lardizabal 4, E-20018 San Sebastián, Spain  2 Université de Strasbourg, CNRS,
Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000
Strasbourg, France  3 Institute for Theoretical Physics, University of
Regensburg, 93053 Regensburg, Germany<EMAIL_ADDRESS>
###### Abstract

In this manuscript, we introduce (symmetric) Tetranacci polynomials $\xi_{j}$
as a twofold generalization of ordinary Tetranacci numbers, considering both
non unity coefficients and generic initial values. We derive a complete closed
form expression for any $\xi_{j}$ with the key feature of a decomposition in
terms of generalized Fibonacci polynomials. For suitable conditions, $\xi_{j}$
can be understood as the superposition of standing waves. The issue of
Tetranacci polynomials originated from their application in condensed matter
physics. We explicitly demonstrate the approach for the spectrum,
eigenvectors, Green’s functions and transmission probability for an atomic
tight binding chain exhibiting both nearest and next nearest neighbor
processes. We demonstrate that in topological trivial models, complex
wavevectors can form bulk states as a result of the open boundary conditions.
We describe how effective next nearest neighbor bonding is engineered in state
of the art theory/ experiment exploiting onsite degrees of freedom and close
range hopping. We argue about experimental tune ability and on-demand complex
wavevectors.

††:

Keywords: Tetranacci polynomials, Fibonacci decomposition, complex
wavevectors, transcendental momentum quantization, bulk-boundary
correspondence, (engineering) next nearest neighbor hopping.

##  1 Introduction

Undoubtedly one of the most famous sequences are the Fibonacci numbers
$f_{n}$, defined recursively by $f_{n+1}=f_{n}+f_{n-1}$ for $n\geq 1$ and
$f_{0}=0,f_{1}=1$ [1, 2, 3, 4]. As noticed by Horadam in the midst of the last
century, generalizations require either altered initial values or
alternatively a modified recursion formula [3]. For instance, Webb and
Parberry did the former and studied Fibonacci polynomials $F_{n}$ obeying
($n\geq 1$) $F_{n+1}=x\,F_{n}+F_{n-1}$, $F_{0}=0,\,F_{1}=1$ [5]. Half a decade
later, Hoggatt Jr. and Long defined generalized Fibonacci polynomials
$\mathcal{F}_{n}$ [6] as ($n\geq 1$)

$\displaystyle\mathcal{F}_{n+1}=x\,\mathcal{F}_{n}+y\,\mathcal{F}_{n-1},\quad\mathcal{F}_{0}=0,\,\mathcal{F}_{1}=1,$
(1)

while Özvatan and Pashaev substituted $\mathcal{F}_{0,1}$ by generic initial
values $G_{0,1}$ [7].

Extending the recursion range in (1) from two to three yields Tribonacci
numbers or Tribonacci polynomials depending on the coefficients and supposing
properly chosen initial values [8, 9, 10, 11, 12]. Subsequently, the first
notion of Tetranacci numbers, where the next element of the sequence is formed
by the previous four, appeared (to our best knowledge) in [8]. Since then,
Tetranacci or Tetranacci-like sequences were studied in many variations up to
modern days, cf. [13, 14, 15, 16, 17] in order to mention only a few. The most
generic form of what we call hereinafter a _Tetranacci polynomials_ $t_{n}$ is
($n\geq 0$)

$\displaystyle
t_{n+2}=x_{1}\,t_{n+1}+x_{0}\,t_{n}+x_{-1}\,t_{n-1}+x_{-2}\,t_{n-2}$ (2)

with some initial values $t_{-2},\ldots,\,t_{1}$ and given coefficients
$x_{1},\ldots,\,x_{-2}$ was previously presented in [14].

In contrast to the generic case, we focus on symmetric Tetranacci polynomials
(cf. (4) below) recovered from (2) for $x_{-2}=-1$ and $x_{1}=x_{-1}$ but
still generic $x_{0,1}$. This particular choice of coefficients seems
arbitrary; it is not. Rather, the eigenvectors of
($a,\,b,\,c,\,\alpha,\,\beta\in\mathbb{R}$)

$\displaystyle M(\alpha,\beta)=\left[\matrix{c-\alpha&b&a\cr b&c&b&a\cr
a&b&c&b&a\cr&\ddots&\ddots&\ddots&\ddots&\ddots\cr&&a&b&c&b&a\cr&&&a&b&c&b\cr&&&&a&b&c-\beta&}ß\right]_{N\times
N}$ (3)

symmetric Toeplitz matrices $M(\alpha=0,\beta=0)$ are composed of symmetric
Tetranacci polynomials as we shall discuss. Finite $\alpha$ or $\beta$ merely
modifies the boundary constraints of eigenvectors from the pure Toeplitz case
[18].

Besides mathematical interests, symmetric Tetranacci appear also in condensed
matter theory. The reason for this is, that we physicists consider most often
particular hermitian systems to which we refer as being ”translation
invariant”. Thus, the model’s physics is captured by (banded) Toeplitz
matrices. For $\alpha=\beta=0$, $M$ mimics an atomic chain with nearest and
next nearest neighbor hopping, while specific $\alpha\neq 0$, $\beta\neq 0$
are connected to the $X-Y$ chain in transverse magnetic field [19] or the
Kitaev chain [18, 19, 20]. It has been shown further, that symmetric
Tetranacci polynomials are linked to characteristic polynomial and the Green’s
functions used in quantum transport [21].

The issue of eigenvalues of banded Toeplitz was investigated formerly in more
generality [22]. Particularly, the spectrum of tridiagonal Toeplitz attracted
some interest [23, 24, 25] as their the eigenvector elements are Chebyshev
[26, 27, 28] or Fibonacci polynomials [29]. Although a matrix as in (3) can be
generated by a product of two tridiagonal Toeplitz matrices [18], results for
the spectrum of $M$ are more complicated w.r.t. to the tridiagonal case as we
shall see.

One of our main contributions is to present a simple and closed form
expression for generic symmetric Tetranacci polynomials, which originates from
a decomposition into generalized Fibonacci polynomials (cf. section 3). As
motivation, the fundamental conviction of physicists is that eigenstates
(eigenvectors) of finite systems are given in terms of standing waves. Their
form is sinusoidal and the perhaps most evident example are oscillations of a
guitar string, being fixed at both ends. By the knowledge that Binet/ Binet-
like forms of Fibonacci polynomials can be reshaped into a sine function [5,
6], the stage was set.

The manuscript is organized as follows. In section 2, we formally define
symmetric Tetranacci polynomials and present the basic strategy to find their
closed form expression. Subsequently, we introduce so called _basic_
Tetranacci polynomials and discuss a few of their properties. In section 3, we
demonstrate that specific generalized Fibonacci polynomials obey also the
Tetranacci recursion formula. We verify that any generic Tetranacci polynomial
can be expressed in terms of those specific solutions. Then, we focus on
applications in quantum physics. In section 4, we discuss the spectrum and the
eigenvectors of an atomic chain owning nearest and next nearest neighbor
hopping granting (3) for $\alpha=\beta=0$. Since some feature of this model
require the next nearest neighbor process to dominate the close-range
coupling, section 5 discusses how arbitrarily large ratios of effective
couplings can be engineered experimentally. We conclude in section 6. A (B)
discusses (states closed formulae for) degenerate roots of the Tetranacci
recursion. Finally, C presents the application to quantum transport. Readers
mainly interested in the application may not hesitate to proceed from section
4 on.

##  2 Generic properties of symmetric Tetranacci polynomials

######  Definition 2.1.

The symmetric Tetranacci polynomial $\xi_{j}$ is recursively defined by

$\displaystyle\xi_{j+2}\,=\,\zeta\,\xi_{j}-\xi_{j-2}\,+\,\eta\left(\xi_{j+1}+\xi_{j-1}\right),\quad
j\in\mathbb{Z}$ (4)

in terms of its initial values $\xi_{i}=g_{i}(\zeta,\,\eta)\in\mathbb{C}$
($i=-2,\,\ldots,\,1$) and complex coefficients $\zeta,\,\eta$.

Although the initial values may or may not depend themselves on $\zeta$ and/
or $\eta$, we utilize always the shorthand notation of $g_{-2},\ldots,\,g_{1}$
and $\xi_{j}$ respectively. For the purpose of illustration, the first few
terms of the sequence read

$\displaystyle\xi_{2}$
$\displaystyle=-g_{-2}+\eta\,g_{-1}+\zeta\,g_{0}+\eta\,g_{1},$ (5a)
$\displaystyle\xi_{3}\,$
$\displaystyle=\,-\eta\,g_{-2}+\left(\eta^{2}-1\right)\,g_{-1}+\eta\left(\zeta+1\right)\,g_{0}+\left(\eta^{2}+\zeta\right)\,g_{1},$
(5b) $\displaystyle\xi_{4}$
$\displaystyle=-\left(\eta^{2}+\zeta\right)\,g_{-2}+\eta\left(\eta^{2}+\zeta-1\right)\,g_{-1}+\left(\zeta+1\right)\left(\zeta-1+\eta^{2}\right)\,g_{0}$
(5c) $\displaystyle\qquad+\eta\left(\eta^{2}+2\zeta+1\right)\,g_{1}$

and further ones follow from (4). Alternatively, we may also rely on the
generating function $E(t)$.

######  Proposition 1.

The generating function $E(t)=\sum\limits_{k=0}^{\infty}\xi_{k}\,t^{k}$ of
symmetric Tetranacci polynomials reads

$\displaystyle
E(t)=\frac{g_{1}\,t\,+\,g_{0}\,(1-\eta\,t)\,+\,g_{-1}\,(\eta\,t^{2}-t^{3})-g_{-2}\,t^{2}}{1-\eta\,t-\zeta\,t^{2}-\eta\,t^{3}+t^{4}}.$
(5f)

######  Proof.

Using the definition of the generating function grants

$\displaystyle E(t)$
$\displaystyle=g_{0}\,+\,g_{1}\,t+t^{2}\,\sum\limits_{n=0}^{\infty}\xi_{n+2}\,t^{n}$
(5g)
$\displaystyle=g_{0}\,+\,g_{1}\,t+t^{2}\,\sum\limits_{n=0}^{\infty}\left[\zeta\,\xi_{n}-\xi_{n-2}+\eta\left(\xi_{n+1}+\xi_{n-1}\right)\right]\,t^{n}$
$\displaystyle=g_{0}\,+\,g_{1}\,t+g_{-1}t^{2}\,(\eta-t)-g_{-2}\,t^{2}-g_{0}\,\eta
t+E(t)\left(\zeta t^{2}-t^{4}+\eta t^{3}+\eta\,t\right).$

Here, we substituted (4) in the last term and all sums were completed properly
in order to provide $E(t)$. $\square$

Perhaps contrary to their appearance in (5a)- (5c), the looked for closed form
of $\xi_{j}$ is rather simple. Since the intention of our mindset is the
applicability, we aim at a particular expression for $\xi_{j}$, namely (5i)
below, demanding the introduction of specific solutions to (4), hereinafter
referred to as the basic Tetranacci polynomials.

######  Definition 2.2.

The basic Tetranacci polynomials $\mathcal{T}_{i}(j)$ ($i=-2,\ldots,\,1$)
satisfy (4) for generic $j\in\mathbb{Z}$. Their initial values are summarized
by

$\displaystyle\mathcal{T}_{i}(j)\,=\,\delta_{ij},\quad i,j=-2,\ldots,\,1.$
(5h)

and $\delta_{ij}$ denotes the Kronecker-Delta111The Kronecker-Delta is defined
as $\delta_{nl}=1$ for $n=l$ and zero otherwise.. We call (5h) the selective
property of $\mathcal{T}_{i}(j)$ as becomes evident hereinafter.

For the purpose of illustration, Table 1 presents the first few terms of
$\mathcal{T}_{i}(j)$. Additional ones can be anticipated from (5a)-(5c). The
primary advantage of the basic Tetranacci polynomial resides in the fact that
the arbitrary initial values of $\xi_{j}$ and the recursion formula (4)
separate by means of the selective property (5h). A similar strategy was
pursued in [16] for Tetranacci numbers. Nevertheless, we then have _initially_
to deal with four symmetric Tetranacci polynomials rather than only one.

######  Corollary 1.

Any symmetric Tetranacci polynomial $\xi_{j}$ can be written as
($j\in\mathbb{Z}$)

$\displaystyle\xi_{j}\,=\,\sum\limits_{i=-2}^{1}\,g_{i}\,\mathcal{T}_{i}(j)$
(5i)

for generic $\eta$, $\zeta\in\mathbb{C}$ and complex initial values
$\xi_{i}=g_{i}$, $i=-2,\ldots,\,1$.

######  Proof.

Due to the linearity of the Tetranacci recursion formula, any linear
combination of solutions also satisfies (4); thus, the l.h.s. of (5i) is a
symmetric Tetranacci polynomial. Hence, in case that (5i) holds already for
the initial values, this relation is true for generic integer $j$. Indeed, we
find that $(j=-2,\,\ldots,\,1)$

$\displaystyle\xi_{j}\,=\,\sum\limits_{i=-2}^{1}\,g_{i}\,\mathcal{T}_{i}(j)\,=\,\,\sum\limits_{i=-2}^{1}\,g_{i}\,\delta_{ij}\,=\,g_{j}$
(5j)

is correct, substituting (5h) in the intermediate step. $\square$

Table 1: Basic Tetranacci polynomials $\mathcal{T}_{i}(j)$ for
$j=-3,\,\ldots,\,2$. The central columns ($j=-2,\,\ldots,\,1$) provide the
intitial values according to (5h). The inversion point ($\bullet$) proposes
the relations $\mathcal{T}_{0}(j)\,=\,\mathcal{T}_{-1}(-1-j)$ and
$\mathcal{T}_{1}(j)\,=\,\mathcal{T}_{-2}(-1-j)$ for arbitrary $j$, $\eta$,
$\zeta$ proven later in Lemma 2.1.

Naturally, the description of $\xi_{j}$ in terms of $\mathcal{T}_{i}(j)$ is
not specific for Tetranacci polynomials and small modifications in both
Definition 2.2 and (5i) extend this strategy to arbitrary (linear) recursive
problems.

The basic Tetranacci polynomials inherit some specific traits originating from
their particular initial values, as can be anticipated from Table 1. More
importantly though, we find interconnections between $\mathcal{T}_{1}$
($\mathcal{T}_{-1}$) and $\mathcal{T}_{-2}$ ($\mathcal{T}_{0}$) reducing
effectively the number of involved quantities. Actually, the Lemmata 2.1, 2.2
below even demonstrate that $\mathcal{T}_{-1}$, $\mathcal{T}_{0}$ and
$\mathcal{T}_{1}$ can be constructed solely from $\mathcal{T}_{-2}$.

######  Lemma 2.1.

The basic Tetranacci polynomials $\mathcal{T}_{i}(j)$ ($i=-2,\,\ldots,\,1$)
obey

$\displaystyle\mathcal{T}_{1}(j)\,=\,\mathcal{T}_{-2}(-1-j),$ (5ka)
$\displaystyle\mathcal{T}_{0}(j)\,=\,\mathcal{T}_{-1}(-1-j),$ (5kb)
$\displaystyle\mathcal{T}_{-2}(j)\,=\,\mathcal{T}_{1}(-1-j),$ (5kc)
$\displaystyle\mathcal{T}_{-1}(j)\,=\,\mathcal{T}_{0}(-1-j),$ (5kd)

for all $j\in\mathbb{Z}$ and generic $\zeta,\,\eta\in\mathbb{C}$.

######  Proof.

Notice that once the validity of (5ka) ((5kb)) is shown, (5kc) ((5kd)) follows
automatically by setting $l=-1-j$ and renaming $l\rightarrow j$ afterwards.
Since the proofs of (5ka), (5kb) are similar, we focus only on the former. The
presented values in Table 1 imply the validity of (5ka) already for
$j=-2,\,-1,\,0$: $\mathcal{T}_{1}(j)=\mathcal{T}_{-2}(-1-j)=0$. At $j=-3$, we
find $\mathcal{T}_{1}(j)=\mathcal{T}_{-2}(-1-j)=-1$. Assuming that (5ka) holds
already for some integers $n-2,\,n-1,\,n,\,n+1$, we are left to demonstrate
(5ka) at $n+2$ ($n-3$) for increasing (decreasing) indices. Since
$\mathcal{T}_{1}(j)$, $\mathcal{T}_{-2}(j)$ are symmetric Tetranacci
polynomials, (4) gives

$\displaystyle\mathcal{T}_{1}(n+2)\,=\,\zeta\,\mathcal{T}_{1}(n)\,-\,\mathcal{T}_{1}(n-2)\,+\,\eta\,\left[\mathcal{T}_{1}(n+1)\,+\,\mathcal{T}_{1}(n-1)\right]$
(5l)

at $j=n$. Similarly at $j=-1-n$, we find

$\displaystyle\quad~{}\mathcal{T}_{-2}(-3-n)\,=\,\zeta\,\mathcal{T}_{-2}(-1-n)\,-\,\mathcal{T}_{-2}(1-n)\,+\,\eta\,\left[\mathcal{T}_{-2}(-2-n)+\mathcal{T}_{-2}(-n)\right],$
(5m)

after reordering the terms. Due to our assumption, we find that (5l), (5m) are
identical which is equivalent to $\mathcal{T}_{1}(j)=\mathcal{T}_{-2}(-1-j)$
at $j=n+2$. The demonstration for decreasing indices, i.e. for $n-3$, is
carried out analogously by exchanging the $j+2$ and $j-2$ terms in (4).
$\square$

######  Lemma 2.2.

The basic Tetranacci polynomials
$\mathcal{T}_{-2}(j),\,\ldots,\,\mathcal{T}_{1}(j)$ obey

$\displaystyle\mathcal{T}_{-2}(j)\,$ $\displaystyle=\,-\mathcal{T}_{-2}(-j),$
(5na) $\displaystyle\mathcal{T}_{-1}(j)\,$
$\displaystyle=\,\mathcal{T}_{-2}(j-1)-\eta\,\mathcal{T}_{-2}(j),$ (5nb)
$\displaystyle\mathcal{T}_{0}(j)\,$
$\displaystyle=\,\eta\,\mathcal{T}_{-2}(j+1)-\mathcal{T}_{-2}(j+2),$ (5nc)
$\displaystyle\mathcal{T}_{1}(j)\,$ $\displaystyle=\,-\mathcal{T}_{-2}(j+1),$
(5nd)

for all $j\in\mathbb{Z}$ and generic $\zeta,\,\eta\in\mathbb{C}$.

######  Proof.

We focus first on (5na), where a proof for $j\geq 0$ is sufficient.
Apparently, (5na) is already valid for $j=0,\,1,\,2$ (cf. Table 1). For $j=3$,
we obtain $\mathcal{T}_{-2}(3)=-\eta$ from (4), i.e. we find
$\mathcal{T}_{-2}(3)=-\eta=-\mathcal{T}_{-2}(-3)$. Assuming (5na) is true for
$n-2,\,n-1,\,n,\,n+1$ ($n\geq 2$) we demonstrate its validity at $n+2$. (4),
implies

$\displaystyle\qquad\mathcal{T}_{-2}(n+2)$
$\displaystyle=\zeta\,\mathcal{T}_{-2}(n)-\mathcal{T}_{-2}(n-2)+\eta\left[\mathcal{T}_{-2}(n+1)+\mathcal{T}_{-2}(n-1)\right],$
(5o) $\displaystyle\qquad\mathcal{T}_{-2}(-n-2)$
$\displaystyle=\zeta\,\mathcal{T}_{-2}(-n)-\mathcal{T}_{-2}(2-n)+\eta\left[\mathcal{T}_{-2}(-1-n)+\mathcal{T}_{-2}(1-n)\right].$
(5p)

Due to our assumption, the two expressions differ only by a sign. Thus, (5na)
is valid. Next, we focus on (5nb). $\mathcal{T}_{-2}(j-1)$ is apparently a
solution to (4), since $\mathcal{T}_{-2}(j)$ is a symmetric Tetranacci
polynomial, i.e. $\mathcal{T}_{-2}(j-1)-\eta\,\mathcal{T}_{-2}(j)$ is one as
well due to the linearity of (4). Hence, the latter has only to satisfy the
selective property of $\mathcal{T}_{-1}(j)$ for (5nb) to be correct. Indeed,
we have (cf. Table 1)

$\displaystyle j=-2:$
$\displaystyle\quad\mathcal{T}_{-2}(-3)-\eta\,\mathcal{T}_{-2}(-2)=\eta-\eta=0\equiv\mathcal{T}_{-1}(-2),$
(5qa) $\displaystyle j=-1:$
$\displaystyle\quad\mathcal{T}_{-2}(-2)-\eta\,\mathcal{T}_{-2}(-1)=1-0=1\equiv\mathcal{T}_{-1}(-1),$
(5qb) $\displaystyle j=0:$
$\displaystyle\quad\mathcal{T}_{-2}(-1)-\eta\,\mathcal{T}_{-2}(0)=0-0=0\equiv\mathcal{T}_{-1}(0),$
(5qc) $\displaystyle j=1:$
$\displaystyle\quad\mathcal{T}_{-2}(0)-\eta\,\mathcal{T}_{-2}(1)=0-0=0\equiv\mathcal{T}_{-1}(1).$
(5qd)

The correctness of (5nc) is a direct consequence of (5kb), (5na), (5nb):

$\displaystyle\mathcal{T}_{0}(j)=\mathcal{T}_{-1}(-1-j)$
$\displaystyle=\mathcal{T}_{-2}(-2-j)-\eta\,\mathcal{T}_{-2}(-1-j)$ (5qr)
$\displaystyle=\eta\,\mathcal{T}_{-2}(j+1)-\mathcal{T}_{-2}(2+j),$

while (5nd) follows from (5ka), (5na):

$\displaystyle\mathcal{T}_{1}(j)=\mathcal{T}_{-2}(-1-j)=-\mathcal{T}_{-2}(j+1).$
(5qs)

Thus, the statements are correct. $\square$

In the view of Corollary 1 and the Lemmata 2.1, 2.2, the closed form
expression of an arbitrary Tetranacci polynomial $\xi_{j}$ demands merely the
one of $\mathcal{T}_{-2}$. Yet the final result for $\mathcal{T}_{-2}$
presented in Theorem 3.2 below requires some preparation.

Furthermore, (5i) is also interesting when studying algebraic properties of
$\xi_{j}$. For instance, we find ($j\in\mathbb{Z}$)

$\displaystyle\xi_{-1-j}=g_{-2}\,\mathcal{T}_{1}(j)+g_{-1}\,\mathcal{T}_{0}(j)+g_{0}\,\mathcal{T}_{-1}(j)+g_{1}\,\mathcal{T}_{-2}(j)$
(5t)

by imposing (5ka)-(5kd) on (5i). Thus, $\xi_{-1-j}=\xi_{j}$ holds in case that
$g_{-2}=g_{1}$ and $g_{-1}=g_{0}$ are true. Similar properties of $\xi_{j}$
may follow, once they have been proven for
$\mathcal{T}_{-2}(j),\,\ldots,\,\mathcal{T}_{1}(j)$.

Although (5u) below is rather trivial from the mathematical point of view,
i.e. that $\xi_{j}$ can be written as linear combination of complex entities,
the Lemma summarizes (to our best knowledge) all relations necessary to
diagonalize symmetric Toeplitz matrices of bandwidth two. As long as (5u) is
valid, this relation is consistent with the in solid state physics famous
Bloch’s theorem without touching further details [30, 31]. Nevertheless, the
consequences imposed by the quantities $S_{1,2}$ (defined in Lemma 2.3 below)
are of fundamental importance for us.

######  Lemma 2.3.

Any symmetric Tetranacci polynomial can be expressed as

$\displaystyle\xi_{j}\,=\,A\,e^{\rmi\theta_{1}\,j}\,+\,B\,e^{-\rmi\theta_{1}\,j}\,+\,C\,e^{\rmi\theta_{2}\,j}\,+\,D\,e^{-\rmi\theta_{2}\,j},$
(5u)

provided that $S_{1}\neq S_{2}$ and $S_{1,2}\neq\pm 2$ hold true where
$S_{1,2}=(\eta\pm\sqrt{\eta^{2}+4(\zeta+2)}\,)/2$. In (5u), we introduced
$\theta_{1,2}\in\mathbb{C}$ defined by $2\cos(\theta_{1,2})=S_{1,2}$. The
coefficients $A$, $B$, $C$, $D$ are set implicitly by $\xi_{i}=g_{i}$,
$i=-2,\ldots,\,1$.

######  Proof.

The announced result is found straightforwardly by the power law ansatz
$\xi_{j}\propto r^{j}$ ($r\neq 0$) on (4). After substituting the ansatz and
dividing by $r^{j}\neq 0$, we arrive at the characteristic equation:

$\displaystyle
r^{2}\,+\,\frac{1}{r^{2}}\,-\,\zeta\,-\,\eta\left(r+\frac{1}{r}\right)\,=\,0.$
(5v)

Its peculiar form suggests to introduce the variable $S=r+r^{-1}$, granting in
turn the quadratic equation $S^{2}-\eta S-\zeta-2=0$, whose zeros are

$\displaystyle S_{1,2}=\frac{\eta\pm\sqrt{\eta^{2}+4(\zeta+2)}}{2}.$ (5w)

Solving $S=r+r^{-1}$ for $r$ at $S=S_{1,2}$ yields

$\displaystyle r_{\pm l}\,=\,\frac{S_{l}\pm\sqrt{S_{l}^{2}-4}}{2},\quad
l=1,\,2,$ (5x)

possessing the properties of $r_{+l}r_{-l}=1$ and $r_{+l}+r_{-l}=S_{l}$ for
$l=1,2$. In case of $S_{1}\neq S_{2}$, $S_{1,2}\neq\pm 2$, we can thus express
$\xi_{j}$ as their linear combination:

$\displaystyle\xi_{j}=A\,r_{+1}^{j}+B\,r_{-1}^{j}+C\,r_{+2}^{j}+D\,r_{-2}^{j}.$
(5y)

The coefficients $A$, $B$, $C$, $D$ are to be set by the initial values
$g_{-2},\ldots,g_{1}$ of $\xi_{j}$. Introducing $\theta_{1,2}$ by
$2\cos(\theta_{1,2})=S_{1,2}$ grants $r_{\pm l}=\exp(\pm\rmi\theta_{l})$.
$\square$

Any degeneracy of the roots $r_{\pm 1,2}$ alters (5u) qualitatively, i.e. the
closed form expression for $\xi_{j}$ will change, and we refer here to A for
further details. Perhaps contrary to the impression of the reader that we
apply next Lemma 2.3 to determine $\mathcal{T}_{-2}$ or $\xi_{-2}$ resulting
in a Binet-like form, similar to the one for Tetranacci numbers in [14], we
follow a different strategy. In fact specific generalized Fibonacci
polynomials also satisfy (4) out of which $\mathcal{T}_{-2}$ can be
constructed. One has still to distinguish the cases of: i) $S_{1}\neq S_{2}$,
ii) $S_{1}=S_{2}$, but $S_{1}^{2}\neq 4$ and iii) $S_{1}=S_{2}$,
$S_{1}^{2}=4$.

##  3 The Fibonacci decomposition

Generalized Fibonacci polynomials, defined here according to (1) (cf. [6]),
are closely related to symmetric Tetranacci polynomials. The trivial limit is
$\eta=0$, where (4) simplifies to $\xi_{j+2}=\zeta\xi_{j}-\xi_{j-2}$; thus,
separating even and odd indices $j$. Defining then $v_{l}=\xi_{2l}$
($u_{l}=\xi_{2l+1}$) immediately grants $u_{l+1}=\zeta u_{l}-u_{l-1}$,
$v_{l+1}=\zeta v_{l}-v_{l-1}$. Yet, even for $\eta\neq 0$ we find that
specific symmetric Tetranacci polynomials obey simultaneously a two term
recursion formula.

######  Theorem 3.1.

The generalized Fibonacci polynomial $\varphi_{l}(j)$ ($l=1,\,2$), set by

$\displaystyle\varphi_{l}(j+1)\,=\,S_{l}\,\varphi_{l}(j)-\,\varphi_{l}(j-1),\quad
j\in\mathbb{Z}$ (5z)

with $S_{1,2}\,=\,(\eta\pm\sqrt{\eta^{2}\,+\,4\left(\zeta+2\right)})/2$ and
initial values $\varphi_{l}(0)=0$, $\varphi_{l}(1)=1$ is a symmetric
Tetranacci polynomial.

######  Proof.

For the sake of clarity, we suppress the index $l=1,2$ in the following. The
proposed statement follows straightforwardly by assuming initially that
$\varphi_{l}(j)$ obeys $\varphi(j+1)\,=\,x\,\varphi(j)+\,y\,\varphi(j-1)$
($j\in\mathbb{Z}$) for arbitrary (complex) $x$, $y$ and initial values
$f_{0}$, $f_{1}$. In order to be a symmetric Tetranacci polynomial,
$\varphi_{l}(j)$ has to satisfy also (4). Replacing in (4) all terms carrying
the indices $j+2$, $j+1$ grant

$\displaystyle(x^{2}+y-\eta\,x-\zeta)\,\varphi(j)\,=\,(\eta y+\eta-
xy)\,\varphi(j-1)\,-\,\varphi(j-2).$ (5aa)

Comparing the coefficients between (5aa) and our ansatz for $\varphi$ sets
$y=-1$ immediately. In turn, $\eta y+\eta-xy=x$ holds without restrictions on
$x$. Instead, the latter is set by $1=x^{2}+y-\eta\,x-\zeta\equiv x^{2}-\eta
x-\zeta-1$ after substituting $y$. The associated quadratic equation has the
two roots $S_{1,2}=(\eta\pm\sqrt{\eta^{2}\,+\,4\left(\zeta+2\right)}\,)/2$
introduced earlier in Lemma 2.3. Thus, $\varphi(j)$ obeys (4) for the
announced coefficients. As $\varphi(j)$ is well defined by $f_{0}$, $f_{1}$
and (5z), the initial values $\varphi(-2)$, $\varphi(-1)$, $\varphi(0)=f_{0}$,
$\varphi(1)=f_{1}$ for the Tetranacci recursion in (4) are fixed. Hence,
$\varphi(j)$ satisfies the definition of symmetric Tetranacci polynomials.
Without loss of generality, we choose $f_{0}=0$ and $f_{1}=1$ for simplicity.
$\square$

Notice that Theorem 3.1 is an implication: An arbitrary symmetric Tetranacci
polynomial $\xi_{j}$ will not obey (5z) due to its generic initial values
$g_{-2},\ldots,g_{1}$. For instance, we may choose
$g_{-2}=\varphi_{1}(-2)+\epsilon$, $g_{-1}=\varphi_{1}(-1)$ and
$g_{0}=\varphi_{1}(0)=0$, $g_{1}=\varphi_{1}(1)=1$ for $\epsilon>0$.

Further, Table 2 exposes the first few values of $\varphi_{1,2}$, from where
we deduce the next proposition before we turn to their closed form expression.

  * $j$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---|---|---|---
$\varphi_{l}(j)$ | $-(S_{l}^{2}-1)$ | $-S_{l}$ | $-1$ | $0$ | $1$ | $S_{l}$ | $S_{l}^{2}-1$

Table 2: The first terms of $\varphi_{l}(j)$ ($l=1,\,2$) for
$j=-3,\,\ldots,3$.

######  Proposition 2.

The polynomials $\varphi_{l}(j)$ ($l=1,2$) satisfy
$\varphi_{l}(j)=-\varphi_{l}(-j)$ for all $j\in\mathbb{Z}$.

######  Proof.

(5z) and the initial values $\varphi_{l}(0)=0$, $\varphi_{l}(1)=1$ yield
$\varphi_{l}(-1)=-1$. Thus, the statement is correct for $j=0,\,1$. Assuming
that $\varphi_{l}(j)=-\varphi_{l}(-j)$ holds already at $n,\,n+1$
($n\in\mathbb{N}_{0}$), (5z) states that

$\displaystyle\varphi_{l}(n+2)\,=\,S_{l}\,\varphi_{l}(n+1)-\,\varphi_{l}(n)\,=\,-\left[S_{l}\varphi_{l}(-n-1)-\varphi_{l}(-n)\right]$
(5ab)

is true. Exchanging $\varphi_{l}(j+1)$, $\varphi_{l}(j-1)$ in (5z) and using
that $n\geq 0$ implies $-n-1<-n$, identifies
$S_{l}\varphi_{l}(-n-1)-\varphi_{l}(-n)=\varphi_{l}(-n-2)$. $\square$

######  Proposition 3.

The explicit closed form expression for $\varphi_{l}(j)$ ($l=1,2$,
$j\in\mathbb{Z}$)

$\displaystyle\varphi_{l}(j)=\frac{r_{+l}^{j}-r_{-l}^{j}}{r_{+l}-r_{-l}}$
(5ac)

is Binet-like whenever $S_{l}\neq\pm 2$. In terms of $\theta_{1,2}$, one has
$\varphi_{l}=\sin(\theta_{l}j)/\sin(\theta_{l})$ [5, 6]. For $S_{l}=\pm 2$, we
find

$\displaystyle\varphi_{l}(j)=j\left(\frac{S_{l}}{2}\right)^{j+1}.$ (5ad)

######  Proof.

First, we focus on $S_{l}\neq\pm 2$. Although (5ac) and its version in terms
of $\theta_{l}$ is known as the closed form expression for generalized
Fibonacci polynomials (cf. [5, 6]), we re-derive it for completeness and in
order to better demonstrate the changes imposed by $S_{l}^{2}=4$ afterwards.
Using the ansatz $\varphi_{l}\propto r^{j}$ ($r\neq 0$) on (5z), we find
$r^{2}-S_{l}\,r+1=0$ after dividing by $r^{j-1}$. The two roots are $r_{\pm
l}=(S_{l}\pm\sqrt{S_{l}^{2}-4})/2$ from (5x) in Lemma 2.3 and $S_{l}\neq\pm 2$
implies $\varphi_{l}(j)=\alpha\,r_{+l}^{j}+\beta\,r_{-l}^{j}$. The
coefficients $\alpha$, $\beta$ are set by $\varphi_{l}(0)=0$ and
$\varphi_{l}(0)=1$ as $\alpha=r_{+l}-r_{-l}$, $\beta=-\alpha$ granting (5ac).
Introducing $\theta_{l}$ as $2\cos(\theta_{l})=S_{l}$ turns (5ac) into
$\sin(\theta_{l}j)/\sin(\theta_{l})$.

For $S_{l}=\pm 2$, the two roots $r_{\pm l}$ become degenerate:
$r_{+l}=r_{-l}=S_{l}/2$. Hence, the linear combination
$\varphi_{l}(j)=\alpha\,r_{+l}^{j}+\beta\,r_{-l}^{j}=\tilde{\alpha}\,r_{+l}^{j}$
becomes insufficient to properly account for the two initial values of
$\varphi_{l}(j)$. Since the recursion formula (5z) does not change
qualitatively at $S_{l}^{2}=4$, one misses in fact the second solution
$j\,r_{+}^{j}$ as demonstrated in the following. Substituting the ansatz
$\varphi_{l}(j)\propto j\,r_{+l}^{j}$ into (5z) and reordering according to
powers in $j$ grants

$\displaystyle j\,(r_{+l}^{2}-S_{l}\,r_{+l}+1)+r_{+l}^{2}-1=0$ (5ae)

where the bracket vanishes since $r_{+l}$ is the root of $r^{2}-S_{l}\,r+1$.
The second term vanishes due to $r_{+l}=S_{l}/2=\pm 1$ found from
$S_{l}^{2}=4$. Thus,
$\varphi_{l}(j)=\tilde{\alpha}\,r_{+l}^{j}+\tilde{\beta}j\,r_{+l}^{j}=(\tilde{\alpha}+\tilde{\beta}\,j)r_{+l}^{j}$
is true. The initial values of $\varphi_{l}(j)$ set $\tilde{\alpha}=0$,
$\tilde{\beta}=1/r_{+}=r_{+}$ and (5ad) is found. $\square$

Similar as for ”ordinary” symmetric Tetranacci polynomials $\xi_{j}$, the
situation of $S_{1}=S_{2}$ offers further special solutions to (4) apart from
only $\varphi_{1,2}(j)$ (cf. Lemma 2.3 and A). The following lemma is the last
intermediate step, before we finally turn to one of the main results of the
article: The decomposition of $\mathcal{T}_{-2}(j)$ (and thus any symmetric
Tetranacci polynomial) in terms of the generalized Fibonacci polynomials
$\varphi_{1,2}(j)$.

######  Lemma 3.1.

For $S_{1}=S_{2}$ (and $S_{1}=\pm 2$) also $j\varphi_{1,2}(j)$
($j^{2}\varphi_{1,2}(j)$) is a symmetric Tetranacci polynomial.

######  Proof.

The situation of $S_{1}=S_{2}$ implies $\varphi_{1}(j)=\varphi_{2}(j)$ for all
$j\in\mathbb{Z}$ (cf. Theorem 3.1) and thus; we demonstrate the statement for
only $j\varphi_{1}(j)$ and $j^{2}\varphi_{1}(j)$. After substituting
$j\varphi_{1}(j)$ into (4), and reordering, we arrive at

$\displaystyle 0$
$\displaystyle=j\left\\{\varphi_{1}(j+2)-\zeta\varphi_{1}(j)+\varphi_{1}(j-2)-\eta\left[\varphi_{1}(j+1)+\varphi_{1}(j-1)\right]\right\\}$
(5af)
$\displaystyle\qquad+2\left[\varphi_{1}(j+2)-\varphi_{1}(j-2)\right]-\eta\left[\varphi_{1}(j+1)-\varphi_{1}(j-1)\right]$
$\displaystyle=2\left[\varphi_{1}(j+2)-\varphi_{1}(j-2)\right]-\eta\left[\varphi_{1}(j+1)-\varphi_{1}(j-1)\right],$

where the curly bracket is identically zero, since $\varphi_{1}(j)$ is a
symmetric Tetranacci polynomial (cf. Theorem 3.1). So far, we have not imposed
$S_{1}=S_{2}$. According to (5w), we find $S_{1}=\eta/2$ and using (5z) twice
shows that (5af) is indeed satisfied

$\displaystyle\qquad\qquad
2\left[\varphi_{1}(j+2)-\varphi_{1}(j-2)\right]-\eta\left[\varphi_{1}(j+1)-\varphi_{1}(j-1)\right]$
$\displaystyle\qquad\qquad=2\underbrace{\left[\varphi_{1}(j+2)\,-S_{1}\,\varphi_{1}(j+1)\right]}_{=-\varphi_{1}(j)}+2\underbrace{\left[S_{1}\,\varphi_{1}(j-1)\,-\varphi_{1}(j-2)\right]}_{=\varphi_{1}(j)}=0,$
(5ag)

i.e. $j\varphi_{1}(j)$ is a symmetric Tetranacci polynomial. Notice that the
constraint $S_{1}=S_{2}$ is essential, as otherwise only $\eta=S_{1}+S_{2}$ is
correct. Then, we may write $S_{1}=S_{2}+\epsilon$ ($\epsilon\neq 0$) and
(5af) is not satisfied.

Next, in order for $j^{2}\varphi_{1}(j)$ to obey (5af) the additional
constraint $S_{1}^{2}=4$ is mandatory. Substituting $j^{2}\varphi_{1}(j)$ into
(4) and reordering the terms afterwards grants

$\displaystyle 0\,$
$\displaystyle=\,j^{2}\left\\{\varphi_{1}(j+2)-\zeta\varphi_{1}(j)+\varphi(j-2)-\eta\left[\varphi_{1}(j-1)+\varphi_{1}(j+1)\right]\right\\}$
(5ah)
$\displaystyle\qquad+2j\left\\{2\left[\varphi_{1}(j+2)-\varphi_{1}(j-2)\right]-\eta\left[\varphi_{1}(j+1)-\varphi_{1}(j-1)\right]\right\\}$
$\displaystyle\qquad+4\,\varphi_{1}(j+2)+4\,\varphi_{1}(j-2)-\eta\,\left[\varphi_{1}(j+1)+\varphi_{1}(j-1)\right]$
$\displaystyle=4\,\varphi_{1}(j+2)+4\,\varphi_{1}(j-2)-\eta\,\left[\varphi_{1}(j+1)+\varphi_{1}(j-1)\right],$

where the first two lines drop since $\varphi_{1}(j)$, $j\varphi_{1}(j)$ are
symmetric Tetranacci polynomials. Due to (5z) and $S_{1}=\eta/2$, we find

$\displaystyle
4\,\varphi_{1}(j+2)+4\,\varphi_{1}(j-2)-\eta\,\left[\varphi_{1}(j+1)+\varphi_{1}(j-1)\right]$
$\displaystyle=4\left[S_{1}\,\varphi_{1}(j+1)-\varphi_{1}(j)\right]+4\left[S_{1}\,\varphi_{1}(j-1)-\varphi_{1}(j)\right]-\eta\,S_{1}\varphi_{1}(j)$
$\displaystyle=2\left[2S_{1}^{2}-S_{1}\frac{\eta}{2}-4\right]\varphi_{1}(j)$
$\displaystyle=2\left[S_{1}^{2}-4\right]\varphi_{1}(j),$ (5ai)

being zero only for $S_{1}^{2}=4$ at generic $j\in\mathbb{Z}$. $\square$

Next, we construct $\mathcal{T}_{-2}$ in terms of $\varphi_{1,2}(j)$ and
$j\varphi_{1,2}(j)$, $j^{2}\varphi_{1,2}(j)$ when the proper conditions are
met.

######  Theorem 3.2.

The closed form expression of $\mathcal{T}_{-2}(j)$ is

$\displaystyle\mathcal{T}_{-2}(j)=\left\\{\matrix{\frac{\varphi_{2}(j)-\varphi_{1}(j)}{S_{1}-S_{2}},&S_{1}\neq
S_{2}\cr\frac{(1-j)\,\varphi_{1}(j+1)\,+\,(1+j)\,\varphi_{1}(j-1)}{S_{1}^{2}-4},&S_{1}=S_{2},S_{1}^{2}\neq
4\cr\frac{S_{1}\,(1-j^{2})}{12}\,\varphi_{1}(j),&S_{1}=S_{2},S_{1}^{2}=4}\right..$
(5aj)

######  Proof.

For $S_{1}\neq S_{2}$, we have that $\varphi_{1,2}(j)$ satisfy (4), while
$j\varphi_{1,2}(j)$, $j^{2}\varphi_{1,2}(j)$ do not (cf. Theorem 3.1 and Lemma
3.1). Since $S_{1}\neq S_{2}$ also implies $\varphi_{1}(j)\neq\varphi_{2}(j)$
(cf. (5z)) for all $j\in\mathbb{Z}\setminus\\{0,\,1\\}$ and due to the
linearity of (4), $[\varphi_{2}(j)-\varphi_{1}(j)]/(S_{1}-S_{2})$ is a non-
trivial solution to (4). Hence, the statement is correct provided that
$[\varphi_{2}(j)-\varphi_{1}(j)]/(S_{1}-S_{2})$ inherits the selective
property of $\mathcal{T}_{-2}(j)$. Using Table 2, we have indeed that

$\displaystyle\qquad j=-2:$
$\displaystyle\quad\frac{\varphi_{2}(-2)-\varphi_{1}(-2)}{S_{1}-S_{2}}\,=\,\frac{-S_{2}-(-S_{1})}{S_{1}-S_{2}}=1\equiv\mathcal{T}_{-2}(-2),$
(5aka) $\displaystyle\qquad j=-1:$
$\displaystyle\quad\frac{\varphi_{2}(-1)-\varphi_{1}(-1)}{S_{1}-S_{2}}\,=\,\frac{-1-(-1)}{S_{1}-S_{2}}=0\equiv\mathcal{T}_{-2}(-1),$
(5akb) $\displaystyle\qquad j=0:$
$\displaystyle\quad\frac{\varphi_{2}(0)-\varphi_{1}(0)}{S_{1}-S_{2}}\,=\,\frac{0-0}{S_{1}-S_{2}}=0\equiv\mathcal{T}_{-2}(0),$
(5akc) $\displaystyle\qquad j=1:$
$\displaystyle\quad\frac{\varphi_{2}(1)-\varphi_{1}(1)}{S_{1}-S_{2}}\,=\,\frac{1-1}{S_{1}-S_{2}}=0\equiv\mathcal{T}_{-2}(1),$
(5akd)

holds true. Turning to the case of $S_{1}=S_{2}$ but $S_{1}^{2}\neq 4$, we
first find $\varphi_{1}(j)=\varphi_{2}(j)$ (cf. (5z)) for all
$j\in\mathbb{Z}$, but also $j\varphi_{1}(j)$ satisfies now (4) due to Lemma
3.1. Since $\varphi_{1}(j\pm 1)$, $(1\pm j)\varphi_{1}(j\pm 1)$ are apparent
solutions of (4), we find that

$\displaystyle\qquad(1-j)\,\varphi_{1}(j+1)\,+\,(1+j)\,\varphi_{1}(j-1)$
$\displaystyle=2\,\varphi_{1}(j+1)-(j+1)\,\varphi_{1}(j+1)$ (5akal)
$\displaystyle\quad+2\,\varphi_{1}(j-1)\,+(j-1)\,\varphi_{1}(j-1)$

is one too. In addition, we have (cf. Table 2)

$\displaystyle\qquad j=-2:$
$\displaystyle\quad\frac{3\,\varphi_{1}(-1)\,-\,\varphi_{1}(-3)}{S_{1}^{2}-4}=\frac{-3+(S_{1}^{2}-1)}{S_{1}^{2}-4}=1\equiv\mathcal{T}_{-2}(-2),$
(5akama) $\displaystyle\qquad j=-1:$
$\displaystyle\quad\frac{2\,\varphi_{1}(0)\,+\,0\,\varphi_{1}(-2)}{S_{1}^{2}-4}=\frac{0+0}{S_{1}^{2}-4}=0\equiv\mathcal{T}_{-2}(-1),$
(5akamb) $\displaystyle\qquad j=0:$
$\displaystyle\quad\frac{\varphi_{1}(1)\,+\,\varphi_{1}(-1)}{S_{1}^{2}-4}=\frac{1-1}{S_{1}^{2}-4}=0\equiv\mathcal{T}_{-2}(0),$
(5akamc) $\displaystyle\qquad j=1:$
$\displaystyle\quad\frac{0\,\varphi_{1}(2)\,+\,2\,\varphi_{1}(0)}{S_{1}^{2}-4}=\frac{0+0}{S_{1}^{2}-4}=0\equiv\mathcal{T}_{-2}(1),$
(5akamd)

i.e. $\mathcal{T}_{-2}(j)$ is properly constructed. Finally, in case of
$S_{1}=S_{2}$ and $S_{1}^{2}=4$ also $j^{2}\varphi_{1}(j)$ satisfies (4)
(Lemma 3.1). Apparently, $(1-j^{2})\varphi_{1}(j)$ vanishes at $j=\pm 1$ and
also for $j=0$ due to $\varphi_{1}(0)=0$. For $j=-2$, we find

$\displaystyle\frac{S_{1}\,(1-4)}{12}\,\varphi_{1}(-2)=\frac{3S_{1}^{2}}{12}=1\equiv\mathcal{T}_{-2}(-2),$
(5akaman)

and the statement is correct. $\square$

Since the expression for $\mathcal{T}_{-2}(j)$ is known to us, we can next
construct the remaining basic Tetranacci polynomials by applying the Lemmata
2.1, 2.2.

######  Proposition 4.

For generic integer $j$, we have

$\displaystyle\mathcal{T}_{-1}(j)\,$
$\displaystyle=\,\frac{\varphi_{1}(j+1)-\varphi_{2}(j+1)+S_{2}\,\varphi_{1}(j)-S_{1}\,\varphi_{2}(j)}{S_{1}-S_{2}},$
(5aoa) $\displaystyle\mathcal{T}_{0}(j)\,$
$\displaystyle=\,\frac{S_{1}\,\varphi_{2}(j+1)-S_{2}\,\varphi_{1}(j+1)+\varphi_{2}(j)-\varphi_{1}(j)}{S_{1}-S_{2}},$
(5aob) $\displaystyle\mathcal{T}_{1}(j)\,$
$\displaystyle=\,\frac{\varphi_{1}(j+1)-\varphi_{2}(j+1)}{S_{1}-S_{2}},$
(5aoc)

supposing here $S_{1}\neq S_{2}$. The results for $S_{1}=S_{2}$ are presented
in B.

######  Proof.

The displayed formulae follow directly by substituting (5aj) into the
relations from Lemmata 2.1, 2.2 and exploiting the properties of
$\varphi_{1,2}(j)$ drawn in Proposition 2 and Theorem 3.1. Alternatively, the
(5aoa) - (5aoc) are apparently linear combinations of solutions to the
recursion formula in (4) and one is left to demonstrate the respective
selective property, which we delegate as exercise to the reader. $\square$

In the beginning of this manuscript, we promised to provide a rather simple
closed form expression for $\xi_{j}$. On first glance of $\mathcal{T}_{i}(j)$
($i=-2,\,\ldots,\,1$) in Theorem 3.2 and Proposition 4, this seems wrong.
However, substituting the basic Tetranacci polynomials into (1) yields
($j\in\mathbb{Z}$)

$\displaystyle\xi_{j}\,$
$\displaystyle=\,\varphi_{2}(j)\,\frac{g_{-2}-S_{1}\,g_{-1}+g_{0}}{S_{1}-S_{2}}-\varphi_{1}(j)\,\frac{g_{-2}-S_{2}\,g_{-1}+g_{0}}{S_{1}-S_{2}}$
(5ap)
$\displaystyle\quad+\varphi_{1}(j+1)\,\frac{g_{-1}-S_{2}\,g_{0}+g_{1}}{S_{1}-S_{2}}-\varphi_{2}(j+1)\,\frac{g_{-1}-S_{1}\,g_{0}+g_{1}}{S_{1}-S_{2}}$

in case of $S_{1}\neq S_{2}$. Similar expressions can be anticipated also for
$S_{1}=S_{2}$, $S_{1}^{2}\neq 4$ and $S_{1}=S_{2}$, $S_{1}^{2}=4$ from B. In
view of Theorem 3.2 and Lemma 2.2, we demonstrated explicitly the
decomposition of a generic symmetric Tetranacci polynomial $\xi_{j}$ in terms
of the generalized Fibonacci polynomials $\varphi_{1,2}$.

The substitution $\varphi_{1,2}(j)=\sin(\theta_{1,2}j)/\sin(\theta_{1,2})$
($S_{1,2}\neq\pm 2$, cf. Proposition 3) shows that $\xi_{j}$ can be seen as
combination of standing waves. Particularly, we replace
$\theta_{1,2}=k_{1,2}d$ in terms of wavevectors $k_{1,2}$ for physical models
with lattice constant $d$. In case a boundary condition is applied, the values
of $k_{1,2}$ become quantized accordingly [18].

However, these wavevectors are generally complex. The reason is that $r_{\pm
l}=\exp(\pm\rmi\theta_{l})$ ($l=1,2$) is the polar form of $r_{\pm l}$ since
$\theta_{l}=R_{l}+\rmi I_{l}$ ($R_{l},\,I_{l}\in\mathbb{R}$) yields $r_{\pm
l}=|r_{\pm l}|\,\exp(\pm\rmi R_{l})$ with $|r_{\pm l}|=\exp(\mp I_{l})$.
Importantly, the potential complex nature of $k_{1,2}$ is independent of the
model’s topological classification and do not necessarily corresponds to edge
modes. We verify the statement in section 4.4 below.

In the context of a tight binding model, we discuss the potential of symmetric
Tetranacci polynomials in physics. During our in-depth analysis we also show
current limitations of the approach as for instance, unknown non-linear
identities (if existent) of $\mathcal{T}_{i}(j)$.

##  4 Lattice model with nearest and next nearest neighbor hopping

The simplest physical model featuring Tetranacci polynomials as defined in (4)
is a tight binding chain of atoms interconnected by nearest and next nearest
neighbor hopping. In terms of spinless fermionic creation/ annihilation
operators $c^{\dagger}_{j}$, $c_{j}$, the one-dimensional lattice Hamiltonian
reads ($\mu,\,t_{1,2}\in\mathbb{R}$)

$\displaystyle\hat{H}=-\mu\sum\limits_{j=1}^{N}c_{j}^{\dagger}c_{j}-\left[\sum\limits_{n=1}^{2}t_{n}\sum\limits_{j=1}^{N-n}(c_{j+n}^{\dagger}c_{j}+c_{j}^{\dagger}c_{j+n})\right]$
(5aq)

where $\mu$ denotes an onsite energy and ($t_{2}$) $t_{1}$ abbreviates (next)
nearest neighbor coupling. Figure 1 provides a sketch of the model for six
atoms. Contrary to physical intuition, we also allow for $|t_{2}|>|t_{1}|$.
Later in section 5, we discuss how effective $t_{2}$, $t_{1}$ couplings of
arbitrary ratio may be engineered in experiments.

###  4.1 General approach to the spectrum

On the assumptions of both open boundary conditions and finite length, direct
diagonalization methods can be applied on (5aq). The fermionic field
$\hat{\Psi}=(c_{1},\ldots,\,c_{N})$,
$\hat{\Psi}^{\dagger}=(c_{1}^{\dagger},\ldots,\,c_{N}^{\dagger})^{\mathrm{T}}$
yields the hermitian Toeplitz matrix

$\displaystyle\mathcal{H}=\left[\matrix{-\mu&-t_{1}&-t_{2}\cr-
t_{1}&-\mu&-t_{1}&-t_{2}\cr-
t_{2}&-t_{1}&-\mu&-t_{1}&-t_{2}\cr&\ddots&\ddots&\ddots&\ddots&\ddots\cr&&-t_{2}&-t_{1}&-\mu&-t_{1}&-t_{2}\cr&&&-t_{2}&-t_{1}&-\mu&-t_{1}\cr&&&&-t_{2}&-t_{1}&-\mu&}ß\right]_{N\times
N}$ (5ar)

satisfying $\hat{H}=\hat{\Psi}^{\dagger}\mathcal{H}\hat{\Psi}$. Please notice
that $\mathcal{H}$ recovers $M(\alpha=0,\,\beta=0)$ from (3). On first glance
an eigenvector $\vec{\psi}_{E}=\left(\xi_{1},\ldots,\,\xi_{N}\right)$ of (5ar)
to eigenvalue $E\in\mathbb{R}$ obeys ($j=3,\ldots,\,N-2$)

$\displaystyle
0=(E+\mu)\,\xi_{j}+t_{1}\,(\xi_{j+1}+\xi_{j-1})+t_{2}\,(\xi_{j+2}+\xi_{j-2}),$
(5as)

and four boundary conditions

$\displaystyle 0=$
$\displaystyle(E+\mu)\,\xi_{1}+t_{1}\,\xi_{2}+t_{2}\,\xi_{3},$ (5ata)
$\displaystyle 0=$
$\displaystyle(E+\mu)\,\xi_{2}+t_{1}\,(\xi_{3}+\xi_{1})+t_{2}\,\xi_{4},$
(5atb) $\displaystyle 0=$
$\displaystyle(E+\mu)\,\xi_{N-1}+t_{1}\,(\xi_{N-2}+\xi_{N})+t_{2}\,\xi_{N-3},$
(5atc) $\displaystyle 0=$
$\displaystyle(E+\mu)\,\xi_{N}+t_{1}\,\xi_{N-1}+t_{2}\,\xi_{N-2}.$ (5atd)

Since the latter four mimic (5as) apart from missing terms, we define
$\xi_{0},\xi_{-1},\xi_{-2}\ldots$, $\xi_{N},\xi_{N+1},\xi_{N+2}\ldots$ as the
continuation of $\xi_{1},\ldots,\,\xi_{N}$ while the eigenvector
$\vec{\psi}_{E}$ remains untouched. Thus, we establish the Tetranacci
recursion formula ($j\in\mathbb{Z}$)

$\displaystyle\xi_{j+1}=\zeta\,\xi_{j}-\xi_{j-2}+\eta\,\left(\xi_{j+1}+\xi_{j-1}\right),\quad\zeta=-\frac{E+\mu}{t_{2}},\,\eta=-\frac{t_{1}}{t_{2}}$
(5atau)

and the boundary conditions reduce to $\xi_{-1}=\xi_{0}=\xi_{N+1}=\xi_{N+2}=0$
as expected from the physical perspective. Please notice though that we aim
mainly on the situation of $t_{2}\neq 0$, as otherwise (5ar) becomes
tridiagonal corresponding to Fibonacci polynomials [24, 27].

Figure 1: Atomic chain with nearest $t_{1}$ and next nearest neibhor hopping
$t_{2}$.

Next, we provide the connection between (5atau) and physical quantities such
as the bulk dispersion relation and wave vectors $k_{1,2}$. Applying Lemma 2.3
and renaming $\theta_{1,2}\rightarrow k_{1,2}d$ ($d$ is the inter atomic
distance) sets
$\xi_{j}=A\exp(\mathrm{i}k_{1}d_{j})+B\exp(-\mathrm{i}k_{1}d_{j})+C\exp(\mathrm{i}k_{2}d_{j})+D\exp(-\mathrm{i}k_{2}d_{j})$
in terms of two right and two left moving plane waves. Naturally, $k_{1,2}$
are related to the eigenvalue $E\equiv E(k_{1,2})$ and
$E(k_{1})=E(-k_{1})=E(k_{2})=E(-k_{2})$ has to be true. The key relations to
remind are i) $S_{1,2}=r_{\pm 1,2}+r_{\pm 1,2}^{-1}$ and ii) $S^{2}_{1,2}-\eta
S_{1,2}-\zeta-2=0$ relating $k_{1,2}$ and $E$ (kept inside $\zeta$). Indeed,
solving for $E$ yields

$\displaystyle E(k)=-\mu-2t_{1}\cos(kd)-2t_{2}\cos(2kd)$ (5atav)

the bulk dispersion relation at $k=k_{1,2}$ [31]. Usually, (5atav) is found
from (5aq) for $N\rightarrow\infty$ and periodic boundary conditions after a
Fourier analysis. We are left to demonstrate that $E(\pm k_{1})=E(\pm k_{2})$
is true. From (5w), we find $S_{1}+S_{2}=\eta$ granting $S^{2}_{1}-\eta
S_{1}=S_{1}^{2}-(S_{1}+S_{2})S_{1}=-S_{1}S_{2}=S_{2}^{2}-(S_{1}+S_{2})S_{2}=S^{2}_{2}-\eta
S_{2}$. In turn, ii) implies $\zeta=S_{1}^{2}-\eta S_{1}-2=S_{2}^{2}-\eta
S_{2}-2$; thus we have $E(\pm k_{1})=E(\pm k_{2})$. Since the parameters
$\mu$, $t_{1,2}$ and $E$ have the physical dimension of an energy, we refer to
$S_{1}+S_{2}=\eta$ as the equal energy constraint. Written explicitly, we have

$\displaystyle\cos(k_{1}d)+\cos(k_{2}d)=-\frac{t_{1}}{2t_{2}}.$ (5ataw)

By (5atav), (5ataw), the eigenvalues $E$ demand the determination of
$k_{1,2}$. Imposing the open boundary conditions on
$\xi_{j}=A\exp(\mathrm{i}k_{1}d_{j})+B\exp(-\mathrm{i}k_{1}d_{j})+C\exp(\mathrm{i}k_{2}d_{j})+D\exp(-\mathrm{i}k_{2}d_{j})$
grants a homogeneous $4\times 4$ system of equations $M\vec{x}=0$ in
$\vec{x}=(A,\,B,\,C,\,D)^{\mathrm{T}}$. Since $\vec{x}=\vec{0}$ yields
$\vec{\psi}_{E}=\vec{0}$, $M$ is singular. After some algebra, we find
($2k_{\pm}=k_{1}\pm k_{2}$)

$\displaystyle\frac{\sin^{2}\left[k_{+}d\left(N+2\right)\right]}{\sin^{2}(k_{+}d)}=\frac{\sin^{2}\left[k_{-}d\left(N+2\right)\right]}{\sin^{2}(k_{-}d)}$
(5atax)

setting the discrete (quantized) values of $k_{1,2}$. Notice that (5atax) has
to be solved together with (5ataw). Thus, $k_{1,2}$ depends on $t_{1}/t_{2}$.
Due to the matrix size, we expect $N$ pairs $(k_{1},\,k_{2})$ satisfying
(5atax) and without loss of generality, their real part is restrained to the
first Brillouin zone, i.e. $-\pi/d\leq\mathrm{Re}(k_{1,2})\leq\pi/d$, due to
$2\pi$-periodicity of (5ataw), (5atax).

Figure 2: Numerical eigenvalues of (5ar) in terms of $\zeta$, $\eta$ for
$N=20$ ($N=21$) in (a) ((b)). (a) The spectrum features several degeneracies
associated to line crossings at specific parameter values. (b) For $N$ odd, no
degenerate eigenvalues appear at $\eta=0$ (red vertical line). Generally, the
spectrum is symmetric in $\eta$. (c) ((d)) Zoom of (a) ((b)).

Unfortunately though, the transcendental character of (5atax) denies further
progress apart from limiting cases. The appearance of $N+2$ pronounces the
next nearest neighbor character of the model and naive expectations as
$k=n\pi/(N+1)$ ($n=1,\ldots,\,N$) clearly fail. Even worse, $t_{1,2}$ have a
strong influence. For simplicity we discuss the situation of $t_{1}=0$, when
only next nearest neighbor hopping remains. Then, the Hamiltonian (5aq)
possess two separated sublattices consisting of only even (odd) $j$ such that
(5ar) becomes block diagonal in the associated basis. Each block itself is
tridiagonal and mimics a nearest neighbor chain with hopping constant $t_{2}$.
For $N$ even, each eigenvalue is twice degenerated

$\displaystyle E=-\mu-2t_{2}\cos\left(\frac{2n\pi}{N+2}\right),\quad
n=1,\ldots,N/2\quad$ (5atay)

while $N$ odd gives

$\displaystyle E$
$\displaystyle=-\mu-2t_{2}\cos\left(\frac{2n\pi}{N+3}\right),\quad
n=1,\ldots(N+1)/2$ (5ataz) $\displaystyle E$
$\displaystyle=-\mu-2t_{2}\cos\left(\frac{2n\pi}{N+1}\right),\quad
n=1,\ldots(N-1)/2$ (5atba)

and the general dependence in $N$ is non trivial [31]. In (5atay) - (5atba),
we extract $k_{1}d$ from the cosine functions. We have $k_{2}=k_{1}-\pi/d$
corresponding to $k_{+}=k_{1}+\pi/(2d)$, $k_{-}=\pi/(2d)$ and both (5ataw)
(5atax) are satisfied. In Figure 2, the solutions from (5atay) mark the line
closings on the vertical $\eta=0$ axis.

Next, we turn to degenerate eigenvalues. As can be seen from Figure 2, they
exist only for well defined ratios $t_{1}/t_{2}$. We shall derive the exact
conditions (cf. (5atbc) below) and proof that only twofold degeneracies exist.

###  4.2 Degenerate eigenvalues

Initially, we assume that the degeneracy is $D$-fold associated to linear
independent eigenvectors
$\vec{\psi}_{E}^{(d)}=(\xi^{(d)}_{1},\ldots,\xi^{(d)}_{N})$
($d=1,\ldots,\,D\geq 2$) of (5ar). From Corollary (1), we have
$\xi^{(d)}_{j}=\sum_{i=-2}^{1}\,g_{i}^{(d)}\,\mathcal{T}_{i}(j)$ with initial
values $g_{i}^{(d)}$. Notice that the basic Tetranacci polynomials
$\mathcal{T}_{i}(j)$ are the same for all $\vec{\psi}_{E}^{(d)}$. The boundary
condition demands $\xi^{(d)}_{-1}=\xi^{(d)}_{0}=0$ granting
$\xi^{(d)}_{j}=g_{-2}^{(d)}\,\mathcal{T}_{-2}(j)+g_{1}^{(d)}\,\mathcal{T}_{1}(j)$.
Since only two initial values remain, the degeneracy is at best twofold as we
shall see. Due to the linearity of the eigenvector equation and the Tetranacci
recursion formula, linear combinations of eigenvectors correspond to one of
initial values. Without loss of generality, we set
$g_{-2}^{(1)}=g_{1}^{(2)}=0$. Then, we observe that
$\vec{\psi}_{E}^{(3)},\ldots,\,\vec{\psi}_{E}^{(D)}$ are composed of
$\vec{\psi}_{E}^{(1)}$, $\vec{\psi}_{E}^{(2)}$, i.e. $D=2$. Next, we derive
the parameter constraints.

The full boundary condition was not yet imposed on $\vec{\psi}_{E}^{(1)}$,
$\vec{\psi}_{E}^{(2)}$. Demanding $\xi_{N+1}^{(1,2)}=\xi_{N+2}^{(1,2)}=0$,
yields the four constraints
$\mathcal{T}_{1}(N+1)=\mathcal{T}_{1}(N+2)=\mathcal{T}_{-2}(N+1)=\mathcal{T}_{-2}(N+2)=0$
due to $g_{1}^{(1)}\neq 0$, $g_{-2}^{(2)}\neq 0$. Exploiting (5nd), i.e.
$\mathcal{T}_{1}(j)\,=\,-\mathcal{T}_{-2}(j+1)$ grants $\mathcal{T}_{-2}(l)=0$
at $l=N+1,\,N+2,\,N+3$. In physics, we trust in Bloch’s theorem, i.e. only the
case $S_{1}\neq S_{2}$ can be relevant. We verify this assumption later by
counting all crossings in order to ensure that none is missing. From (5ac), we
have $\varphi_{1,2}(N+2)=\sin[k_{1,2}d(N+2)]=0$. Seemingly, one can choose
($n_{1,2}=1,\ldots,\,N+1$)

$\displaystyle k_{1,2}d=\frac{n_{1,2}\pi}{N+2}$ (5atbb)

independently. However, any linear combination of $\vec{\psi}_{E}^{(1)}$,
$\vec{\psi}_{E}^{(2)}$ has to satisfy the boundary conditions. Hence, (5atax)
has to be satisfied as well, imposing selection rules on $k_{1,2}$. In terms
of $n_{\rm max}=(N+2)/2$ ($n_{\rm max}=(N+1)/2$) for even (odd) $N$, we arrive
at ($n=2,\ldots,\,n_{\rm max},\,l=1,\,\ldots,\,n-1$)

$\displaystyle\left(k_{+}d,\,k_{-}d\right)$
$\displaystyle=\left(\frac{n\pi}{N+2},\,\frac{l\pi}{N+2}\right).$ (5atbc)

The ratio for $t_{1}/t_{2}$ is set by (5ataw) upon inserting $k_{\pm}$. Then,
the eigenvalue follows from (5atav). Notice that (5atbc) corresponds to values
of $\eta\geq 0$ ($t_{1}/t_{2}<0$). For $\eta\leq 0$, set
$k_{+}d\rightarrow\pi-k_{+}d$ while $k_{-}d$ is kept fixed. Avoiding double
counting at $\eta=0$ for even $N$, the total number of line crossings is
$N^{2}/4$ (($N^{2}-1)/4$) for $N$ even (odd) in agreement with the numerical
investigation. Henceforth, the earlier restriction on $S_{1}\neq S_{2}$ was
correct indeed. Next, we turn to the eigenvectors.

###  4.3 Eigenvectors and spatial inversion symmetry

Although we ”determine” all eigenvectors in the following, the lack of non-
linear identities (if existent) for generic $\xi_{j}$ or at least for
$\mathcal{T}_{i}(j)$ is a current limitation of the approach and the solution
for $\vec{\psi}_{E}$ appears rather unsatisfying. A similar issue may
potentially arise in quantum transport depending specifically on the
investigated model (cf. C).

When an eigenvalue $E$ is chosen, the eigenvectors are set by the initial
values $g_{-2},\ldots,\,g_{1}$. Imposing the open boundary conditions on (5i)
gives $\xi_{j}=g_{-2}\,\mathcal{T}_{-2}(j)+g_{1}\,\mathcal{T}_{1}(j)$ for non
degenerate eigenvalues. Otherwise, we have
$\xi^{(d)}_{j}=g_{-2}^{(d)}\,\mathcal{T}_{-2}(j)+g_{1}^{(d)}\,\mathcal{T}_{1}(j)$,
$d=1,2$ where $g_{-2}^{(d)}$, $g_{1}^{(d)}$ can be chosen independently.
Therefore, we have

$\displaystyle\xi^{(1)}_{j}=g_{-2}^{(1)}\,\mathcal{T}_{-2}(j),$ (5atbd)
$\displaystyle\xi^{(2)}_{j}=-g_{1}^{(2)}\,\mathcal{T}_{-2}(j+1)$ (5atbe)

exploiting (5nd). Here, $g_{-2}^{(1)}$, $g_{1}^{(2)}$ adopt the role of
normalization constants.

Figure 3: Relation between inversion symmetry $\lambda_{\mathcal{I}}=\pm 1$
and branch $s_{\rm Q}=\pm 1$ in (a), (b) ((c), (d)) for even (odd) $N$.
Numerically, we conclude $\lambda_{\mathcal{I}}\,s_{\rm Q}=-1$ always, i.e.
for all eigenvalues and independently of $N$.

Contrary for the non degenerate case, $\xi_{N+1}=0$ imposes that
$g_{1}=-g_{-2}\mathcal{T}_{-2}(N+1)/\mathcal{T}_{-2}(N+2)$ using (5nd). Both
$\mathcal{T}_{-2}(N+1)$, $\mathcal{T}_{-2}(N+2)$ are finite in the non-
degenerate case. Replacing $g_{1}$, yields

$\displaystyle\xi_{j}=\frac{g_{-2}}{\mathcal{T}_{-2}(N+2)}~{}\left[\mathcal{T}_{-2}(j)\,\mathcal{T}_{-2}(N+2)-\mathcal{T}_{-2}(N+1)\,\mathcal{T}_{-2}(j+1)\right].$
(5atbf)

satisfying the open boundary conditions by construction. Notice though that
$\xi_{N+2}=0$ yields the quantization condition (5atax). Unfortunately,
further progress beyond (5atbf) is denied due to our lack of non-linear
identities for $\mathcal{T}_{-2}(j)$.

The Hamiltonian (5aq) possess time reversal $\tau$ and spatial inversion
symmetry $\mathcal{I}$. Both imply that wavevectors appear in $\pm k$ pairs.
Yet, the attempt of exploiting those fails. Explicitly, we have
$\tau=\mathds{1}_{N}\hat{\mathcal{K}}$ ($\hat{\mathcal{K}}$ denotes the
operator of complex conjugation) for spinless electrons [32] and

$\displaystyle\mathcal{I}=\left[\matrix{&&&&1\cr&&&1\cr&&\iddots\cr&1\cr
1}ß\right]_{N\times N}.$ (5atbg)

Time reversal symmetry has little impact since all quantities in Eq. (5atbf)
are real and $g_{-2}$ can be chosen freely. The inversion symmetry is self-
inverse $\mathcal{I}^{2}=\mathds{1}_{N}$ with eigenvalues
$\lambda_{\mathcal{I}}=\pm 1$. Since
$\mathcal{I}\,\mathcal{H}\,\mathcal{I}=\mathcal{H}$ is true, the eigenvectors
$\vec{\psi}_{E}$ fall into two categories
$\mathcal{I}\,\vec{\psi}_{E}=\lambda_{\mathcal{I}}\vec{\psi}_{E}$. Even (odd)
eigenvectors $\lambda_{\mathcal{I}}=+1$ ($\lambda_{\mathcal{I}}=-1$) obey
$\xi_{N+1-j}=\lambda_{\mathcal{I}}\,\xi_{j}$.

However, this does not solve the basic problem as $\lambda_{\mathcal{I}}$ is
imprinted into the quantization constraint. In fact, (5atax) has two branches
$f(k_{+})=s_{\rm Q}f(k_{-})$ ($f(k)=\sin[k(N+2)]/\sin(k)$) with $s_{\rm Q}=\pm
1$. Numerically, we find $s_{\rm Q}\lambda_{\mathcal{I}}=-1$ as shown in
Figure 3.

The numerical investigation supports the criterion of degenerate energies.
Figure 3 illustrates that line crossings occur between eigenvectors of
opposite spatial inversion symmetry. Henceforth, the degenerate case demands
$f(k_{+})=f(k_{-})$, $f(k_{+})=-f(k_{-})$ simultaneously, i.e.
$f(k_{+})=f(k_{-})=0$ is the only solution. Indeed, both sides of (5atax)
vanish independently upon inserting (5atbc).

###  4.4 The Tetranacci arrow and complex wavevectors

The presence of only $\mathcal{I}$, $\tau$ implies that (5aq) is of class AI,
i.e. topological trivial according to the classification scheme of Altand and
Zirnbauer [33]. Thus, we neither expect the existence of complex wavevectors
nor of topologically protected edgestates. However, complex values of
$k_{1,2}$ are necessary in order to account properly for $t_{2}\rightarrow 0$.
The spectrum effectively separates into two entities associated with only real
wavevectors or alternatively one complex and one real wavevector. In Figure 4
(a), the former is depicted in red. We refer to this region as the Tetranacci
arrow.

Figure 4: Numerical spectrum of (5ar) for $N=40$. (a) Real values of both
$k_{1,2}$ are limited to the Tetranacci arrow in red. Outside and shown in
blue, one wavevector becomes complex. (b) Maximal deviation of eigenvector
entries for $t_{2}$ and $t_{2}\neq 0$. The complex solution for $k$ enables
the proper limit on the spectrum for $\eta\rightarrow\infty$
($t_{2}\rightarrow 0$). Details are stated in the main text.

Naturally, we expect a continuous transition between $|t_{2}|\gg|t_{1}|$ and
$|t_{2}|\ll|t_{1}|$. At $t_{2}=0$ the spectrum of (5ar) is
$E=-\mu-2t_{1}\cos(kd)$ with $kd=n\pi/(N+1)$, $n=1,\ldots,\,N$ [23, 24, 25,
28, 29] setting the tendency of the blue lines in Figure 4 (a). Yet, both
$t_{1,2}$ influence $k_{1,2}$, particularly close to the arrow’s boundary.
Here, (5ataw) suggests complex solutions of $k_{1,2}$ being apparent for
$|t_{1}|>4|t_{2}|$. In Figure 4 (a), the red (blue) color indicates two real
(one complex and one real) wavevector.

Analytically, the boundary conditions (5ata) - (5atd) explain the behavior.
While $t_{1,2}\neq 0$ corresponds to four constraints and two left/ right
mover contributions to $\xi_{j}$, the situation of $t_{2}=0$ implies
$\xi_{0}=\xi_{N+1}=0$ and $\xi_{j}=A\exp(\rmi kdj)+B\exp(-\rmi kdj)$. In order
to satisfy the extreme case of $t_{2}\rightarrow 0$, one wavevector becomes
complex. With increasing imaginary part, the complex wavevectors contribution
to $\xi_{j}$ diminishes as shown in Fig. 4 (b). For the numerical data, we
diagonalized (5ar) at $t_{2}\neq 0$, $t_{2}=0$, ordering eigenvectors from
lowest to largest eigenvalues. The color indicates the largest absolute
deviation of the corresponding normalized eigenvector entries. Of course, the
comparison is fruitless within the Tetranacci arrow where $t_{2}$ dominates. A
closer numerical investigation reveals that the difference diminishes on
exponential scales (as expected from (5u)) and does not change abruptly to
zero outside the arrow. Additionally, non of the eigenvectors of (5ar) is an
edgestate in agreement with the bulk edge correspondence [34, 35].

Despite the complex wavevector outside the Tetranacci arrow, all eigenvalues
lie correctly within the frame set by the bulk dispersion relation and the
first Brillouin zone. Recalling $S_{1,2}^{2}-\eta S_{1,2}-\zeta-2=0$ from
Lemma 2.3 and $S_{1,2}=2\cos(k_{1,2}d)$ provides a direct link to Figure 2.
Due to the $2\pi$ periodicity, the (real part of) $k_{1,2}$ lies between
$-\pi/d$ and $\pi/d$. The two extremes provide the boundaries $\zeta=2\mp
2\eta$ of the blue double fan from Figure 4 (a). In $\zeta-\eta$ space, both
lines cross in the upper tip of the arrow at $(\eta,\,\zeta)=(0,\,2)$. The two
flanks follow $\zeta=2\mp 2\eta$ when $\zeta\leq 2$. For the lower boundary of
the Tetranacci arrow, we notice that $S_{1,2}^{2}-\eta S_{1,2}-\zeta-2=0$
yields smaller values of $\zeta$ when $|S_{1,2}|<2$. The lowest parameterized
curve follows from $\partial\zeta/\partial S_{1,2}=0$ for fixed $\eta$. For
$S_{1,2}=\eta/2$, we have $\zeta=-2-\eta^{2}/4$ applicable only for
$|\eta|\leq 4$ since $k_{1,2}$ are found real and reside within the first
Brillouin zone. For both finite number of atoms $N$ and open boundary
conditions, $\zeta\neq-2-\eta^{2}/4$ as otherwise $S_{1}=S_{2}$ contradicts
Bloch’s theorem by (5w).

##  5 Engineering effective next nearest neighbor coupling

Unfortunately the most interesting part of the model, i.e. the Tetranacci
arrow, is hardly accessible in experiments. However, the basic ingredients of
nearest neighbor processes and onsite degrees of freedom allow the engineering
of effective (next) nearest neighbor bonds $\tilde{t}_{1}$ ($\tilde{t}_{2}$).
Importantly, the ratio $\tilde{t}_{1}/\tilde{t}_{2}$ is easy to manipulate and
tuning it yields arbitrarily large or small values. For the proof of
principle, we consider the $X-Y$ chain in transverse magnetic field and the
Kitaev chain.

Figure 5: Numerical spectrum of the Kitaev chain in terms of $\zeta$, $\eta$
for $N=20$ and $t/\Delta=4$ ($t/\Delta=0.25$) in (a) ((b)). The spectrum
differs for $|t/\Delta|>1$ and $|t/\Delta|<1$ associated to different limiting
cases for $-\tilde{t}_{1}/\tilde{t}_{2}\rightarrow 0$. The Majorana fermions
correspond to the lowest parabolic-like curve around $\eta=0$ [18].

The former consists of $N$ spins placed along a $1d$ axis and neighbors
interact via their $x$ ($y$) components $S_{j}^{\mathrm{x}}$
($S_{j}^{\mathrm{y}}$) [36]. Adding a transverse magnetic field $h$, the model
reads [19]

$\displaystyle\hat{H}_{\mathrm{XY}}=-h\sum\limits_{j=1}^{N}S_{j}^{2}-\sum\limits_{j=1}^{N-1}\left(J_{\mathrm{x}}S_{j}^{\mathrm{x}}S_{j+1}^{\mathrm{x}}+J_{\mathrm{y}}S_{j}^{\mathrm{y}}S_{j+1}^{\mathrm{y}}\right)$
(5atbh)

where $J_{\mathrm{x}}$, $J_{\mathrm{y}}$ abbreviate interaction constants. The
diagonalization of (5atbh) was undertaken in [19] ([36]) for $h\neq 0$ ($h=0$)
using the Jordan-Wigner transformation which replaces spin operators by spin-
less fermionic operators $c_{j}$, $c_{j}^{\dagger}$. The spin chain maps on
the Kitaev chain as pointed out by Zvyagin [37]. Therefore, we start directly
from the latter. Spinless electrons experiencing hopping $t$ and $p$-wave
superconductivity between nearest neighboring atoms, setting the Kitaev chain
as [20]

$\displaystyle\hat{H}_{\mathrm{KC}}$
$\displaystyle=-\mu\sum\limits_{j=1}^{N}\left(c_{j}^{\dagger}c_{j}-\frac{1}{2}\right)-t\sum\limits_{j=1}^{N-1}\left(c_{j+1}^{\dagger}c_{j}+c_{j}^{\dagger}c_{j+1}\right)$
(5atbi)
$\displaystyle\quad+\Delta\sum\limits_{j=1}^{N-1}\left(c_{j+1}^{\dagger}c_{j}^{\dagger}+c_{j+1}c_{j}\right).$

We consider $t$, $\Delta$, $\mu\in\mathbb{R}$ and $N$ atoms. Independently of
the $X-Y$-chain, this model has attracted some attention in the past as it is
the archetypal model for topological superconductors. Although $p$-wave
superconductivity is itself rare in nature, several promising platforms were
proposed in the past decade in order to engineer the desired odd parity
superconductivity [38, 39, 40, 41]. However, the hunt for Majorana fermions/
Majorana zero modes is not our motivation to subsequently introduce Majorana
operators [20, 34]

$\displaystyle\left(\matrix{\gamma^{{\color[rgb]{0,0,1}A}}_{j}\cr\gamma^{{\color[rgb]{1,.5,0}B}}_{j}}\right)=\frac{1}{\sqrt{2}}\left(\matrix{1&1\cr-\mathrm{i}&\mathrm{i}}\right)\left(\matrix{c_{j}\cr
c_{j}^{\dagger}}\right),$ (5atbj)

rather the aim for technical simplicity. The advantage is that Majorana
operators treat $t$ and $\Delta$ equally since they own particle and hole
properties simultaneously:
$(\gamma^{{\color[rgb]{0,0,1}A}}_{j})^{\dagger}=\gamma^{{\color[rgb]{0,0,1}A}}_{j}$,
$(\gamma^{{\color[rgb]{1,.5,0}B}}_{j})^{\dagger}=\gamma^{{\color[rgb]{1,.5,0}B}}_{j}$.
Then, (5atbi) becomes [20, 34]

$\displaystyle\hat{H}_{\mathrm{KC}}=-\mathrm{i}\mu\sum\limits_{j=1}^{N}\,\gamma^{{\color[rgb]{0,0,1}A}}_{j}\,\gamma^{{\color[rgb]{1,.5,0}B}}_{j}+\mathrm{i}(\Delta-t)\sum\limits_{j=1}^{N-1}\,\gamma^{{\color[rgb]{0,0,1}A}}_{j}\,\gamma^{{\color[rgb]{1,.5,0}B}}_{j+1}+\mathrm{i}(\Delta+t)\sum\limits_{j=1}^{N-1}\,\gamma^{{\color[rgb]{1,.5,0}B}}_{j}\,\gamma^{{\color[rgb]{0,0,1}A}}_{j+1}$
(5atbk)

Aiming on the spectrum of (5atbk) suggest the introduction of Majorana
sublattices
$\hat{\Psi}_{\alpha}=\left(\gamma^{\alpha}_{1},\,\ldots,\,\gamma^{\alpha}_{N}\right)^{\mathrm{T}}$,
$\alpha={\color[rgb]{0,0,1}A},\,{\color[rgb]{1,.5,0}B}$. In this basis, the
eigenvectors entries are revealed as symmetric Tetranacci polynomials with
coefficients $\zeta=(E^{2}-\mu^{2}-2t^{2}-2\Delta^{2})/(t^{2}-\Delta^{2})$,
$\eta=-2t\mu/(t^{2}-\Delta^{2})$ [18, 31] and $E$ as eigenvalue. Comparing
$\eta=-\tilde{t}_{1}/\tilde{t}_{2}$ with the lattice model from section 4, we
have effective (next) nearest neighbor hopping _processes_
$\tilde{t}_{1}=2t\mu$ ($\tilde{t}_{2}=t^{2}-\Delta^{2}$), ignoring the wrong
physical dimension of $\tilde{t}_{1,2}$. The ratio (value of)
$\tilde{t}_{1}/\tilde{t}_{2}$ ($\eta$) could be tuned by an electrostatic gate
shifting the potential; thus, modifying the onsite energy $\mu$. From [19], we
have $\tilde{t}_{1}=-2h(J_{x}+J_{y})$, $\tilde{t}_{2}=J_{\rm x}J_{\rm y}$ for
the $X-Y$-chain. Henceforth, a small magnetic field in $z$ direction reflects
strong effective next nearest neighbor coupling. We finish this section
discussing the emergence of the Tetranacci sequence and thus the engineering
of $\tilde{t}_{1,2}$.

Figure 6: Kitaev chain in Majorana sublattice basis for $N=7$ atoms. Solid
(dashed) arrows correspond to $\mathrm{i}\mu$ in black,
$a=\mathrm{i}(\Delta-t)$ ($-a$) in purple and $b=\mathrm{i}(\Delta+t)$ (-b) in
blue. Only the horizontal axis has a spatial extend with $d$ as the
interatomic distance. The vertical axis displays onsite degrees of freedoms.

A first impression can be taken from Fig. 5, where the numerically found
spectrum forms the Tetranacci arrow. Notice though that
$\tilde{t}_{1}/\tilde{t}_{2}\rightarrow\infty$ has distinct limits due the
substructure of $\tilde{t}_{2}$. Analytically, one can avoid the cumbersome
algebra from [18] and deduce the proper result pictorially from the system’s
sketch in Figure 6. The diagonalization of (5atbk) implies a disentanglement
of the sublattices; thus projecting the information of sublattice
$\hat{\Psi}_{{\color[rgb]{1,.5,0}B}}$ on $\hat{\Psi}_{{\color[rgb]{0,0,1}A}}$
(or vice versa). This demands merely to identify all non repetitive processes
from some $\gamma_{j}^{{\color[rgb]{0,0,1}A}}$ to some
$\gamma_{l}^{{\color[rgb]{0,0,1}A}}$ ($j,l=1,\ldots,N$) by following the
arrows and multiplying the associated constants. Ignoring initially all
boundary effects, we use $\gamma_{j}^{{\color[rgb]{0,0,1}A}}$ as starting
position as sketched in Figure 6. Spatial inversion symmetry allows to focus
only on forward processes, i.e. on growing $j$. For nearest neighbors
$\gamma_{j}^{{\color[rgb]{0,0,1}A}}\rightarrow\gamma_{j+1}^{{\color[rgb]{0,0,1}A}}$,
we have $\downarrow{\color[rgb]{0,1,1}\nearrow}\equiv-\mathrm{i}\mu\,b$ and
${\color[rgb]{.75,0,.25}\searrow}\,\uparrow\equiv a\,\mathrm{i}\mu$. For
$\gamma_{j}^{{\color[rgb]{0,0,1}A}}\rightarrow\gamma_{j+2}^{{\color[rgb]{0,0,1}A}}$,
we have ${\color[rgb]{.75,0,.25}\searrow}{\color[rgb]{0,1,1}\nearrow}=ab$
only, without repetitive use of effective nearest neighbor bonds. The
effective onsite terms
$\gamma_{j}^{{\color[rgb]{0,0,1}A}}\rightarrow\gamma_{j}^{{\color[rgb]{0,0,1}A}}$
are $\downarrow\uparrow=\mu^{2}$,
${\color[rgb]{.75,0,.25}\searrow}{\color[rgb]{.75,0,.25}\nwarrow}=-a^{2}$
${\color[rgb]{0,1,1}\swarrow}{\color[rgb]{0,1,1}\nearrow}=-b^{2}$; merely a
corrective factor has to be added when the chain has finite length. Collecting
all information and accounting for the backward processes yields
($j,j^{\prime}=1\,,\ldots,\,N$)

$\displaystyle h_{jj^{\prime}}$
$\displaystyle=\left[\mu^{2}-a^{2}(1-\delta_{jN})-b^{2}(1-\delta_{j1})\right]\,\delta_{j,j^{\prime}}\,+\,\mathrm{i}\mu(a-b)\,\left(\delta_{j,j^{\prime}+1}\,+\,\delta_{j+1,j^{\prime}}\right)$
(5atbl)
$\displaystyle\quad+\,ab\,\left(\delta_{j,j^{\prime}+2}\,+\,\delta_{j+2,j^{\prime}}\right).$

The spectrum of (5atbk) is found from
$h\vec{v}_{{\color[rgb]{0,0,1}A}}=E^{2}\vec{v}_{{\color[rgb]{0,0,1}A}}$ with
$(\vec{v}_{{\color[rgb]{0,0,1}A}},\vec{v}_{{\color[rgb]{1,.5,0}B}})$ as
eigenvector of (5atbk). The entries of both sublattice vectors
$\vec{v}_{{{\color[rgb]{0,0,1}A}},{{\color[rgb]{1,.5,0}B}}}$ are symmetric
Tetranacci polynomials with the earlier mentioned coefficients. Although $h$
has the same structure as $M(\alpha\neq 0,\beta\neq 0)$ from (3) (and is thus
not a Toeplitz matrix)
$h\vec{v}_{{\color[rgb]{0,0,1}A}}=E^{2}\vec{v}_{{\color[rgb]{0,0,1}A}}$ can be
treated along the lines of section 4 [18].

##  6 Conclusion

Subsequent to the definition of symmetric Tetranacci polynomials $\xi_{j}$
($j\in\mathbb{Z}$), we gave a closed form expression in terms of basic
Tetranacci polynomials $\mathcal{T}_{i}(j)$ ($i=-2,\ldots,\,1$). Due to their
initial values $\mathcal{T}_{i}(l)=\delta_{il}$ ($l=-2,\,\ldots,1\,$), they
inherit specific traits and interrelations. In turn, $\xi_{j}$ can be
constructed from $\mathcal{T}_{-2}$ alone. We demonstrated in (5aj) the
decomposition of $\mathcal{T}_{-2}(j)$, and thus generic $\xi_{j}$, in terms
of the Fibonacci/ Tetranacci polynomials $\varphi_{1,2}(j)$. The sinusoidal
representation of $\varphi_{1,2}(j)$ enables a standing wave form of $\xi_{j}$
in (5ap); thus reflecting already their potential in condensed matter physics
when appropriate prerequisites are met. We discussed that Tetranacci
polynomials generally allow for complex wavevectors $k_{1,2}$ independently of
a physical model’s topological classification. Contrary to previous works (cf.
[18, 21]), we both generalized and simplified the presented results beyond
$S_{1}\neq S_{2}$. Based on the rigorous mathematical approach, the shown
results are generic and generalize beyond a concrete physical system; thus,
extending earlier limitations.

Besides the theoretical treatment of Tetranacci polynomials, we have shown
their appearance in three physical systems: The $X-Y$ model in transverse
magnetic field, the Kitaev chain and in atomic tight-binding systems owning
nearest and next nearest neighbor hopping. For the latter, we explicitly
demonstrated that Tetranacci polynomials $\xi_{j}$ set the wave function at an
atomic position $j$. Solving the associated eigenvector equation, we derived a
transcendental quantization condition for the wavevectors $k_{1,2}$, cf.
(5atax). Typically, the solution differs from the naively expected particle in
the box behavior, that is $k=n\pi/(N+1)$ ($n=1,\ldots,\,N$) for $N$ atoms. The
three main reasons are, firstly that $k_{1,2}$ depend on the models parameters
by means of the equal energy constraint $E(\pm k_{1})=E(\pm k_{2})$. Secondly,
the quantization constraint captures the next nearest neighbor character of
the model, i.e. $N+2$ rather than $N+1$ appears. Lastly, $k_{1,2}$ are
generally complex. The momentum quantization condition yields a degenerate
spectrum at well defined parameter values. We deduced and stated the
conditions. Spatial inversion symmetry protect the degenerate eigenvalues and
we demonstrated that the spatial parity of the associated eigenstates is
imprinted into the transcendental constraint (5atax).

This proofed fatal since symmetry relation fail to simplify the expression of
eigenvectors beyond a merely formal solution due to unknown non-linear
identities (if existent) of $\mathcal{T}_{-2}(j)$. Quantum transport (cf. C)
suffers the same fate although model specific traits may allow progress in the
linear transport regime as has been demonstrated earlier for the Kitaev chain
[21].

When the Tetranacci recursion is obeyed, the spectrum of quantum devices
features a shape we dubbed as _Tetranacci arrow_. Inside, wavevectors are real
and all degenerated eigenvalues are placed within. Experimental access to the
Tetranacci arrow is seemingly denied, as the next nearest neighbor hopping
$t_{2}$ has to dominate its nearest neighbor cousin $t_{1}$. However, coupled
onsite degrees of freedom and nearest neighbor processes allows to engineer
effective (next) nearest neighbor bonding $\tilde{t}_{1}$ ($\tilde{t}_{2}$)
with arbitrarily large or small ratio $\tilde{t}_{1}/\tilde{t}_{2}$. We
provided a descriptive proof of principle based on the $X-Y$-model and the
Kitaev chain. For the former, we argued that $\tilde{t}_{1}/\tilde{t}_{2}$
depends on the onsite energy $\mu$ and is thus experimentally tune able by an
electronic gate. For the $X-Y$ chain this ratio depends on the external
magnetic field, where small field strength relate to large
$\tilde{t}_{1}/\tilde{t}_{2}$.

Several setups to simulate the Kitaev chain in table-top experiments using
macroscopic elements were proposed and realized [42, 43]. Particularly, the
usage of magnetic spinners to fabricate a macroscopic version of a quantum
ladder provides an mechanical analog of the Kitaev chain with an adjustable
onsite term [44]. Besides, unpaired electrons in carbon ladder polymers
exhibit ferromagnetic and antiferromagnetic interactions such that effective
$\tilde{t}_{1,2}$ bonds are formed [45]. The discussed physics of an atomic
chain featuring nearest and next nearest neighbor hopping is therefore
experimentally accessible and allows the on-demand facilitation of complex
wavevectors. In short terms, the Tetranacci arrow is real indeed.

We gratefully acknowledge financial support from the Deutsche
Forschungsgemeinschaft (SFB 1277 Project B04), the Elitenetzwerk Bayern (IGK
Topological Insulators) and from the French National Research Agency ANR
through project ANR-20-CE30-0028-01. We acknowledge interesting and fruitful
discussion with M. Grifoni, M. Marganska, D. Weinmann and S. Hrdina. Further,
we thank W. Häusler and M. Nieper-Wisskirchen for encouraging us to write this
manuscript. Finally, we are grateful for all suggestions by the referees.

##  Appendix A Degenerated roots $r_{\pm 1,2}$

######  Proposition 5.

Degeneracies of $r_{\pm l}$ grant

$\displaystyle r_{\pm 1}^{j},\,j\,r_{\pm 1}^{j},$ $\displaystyle
S_{1}=S_{2},\,S_{1}^{2}\neq 4,$ (5atbm) $\displaystyle
r_{+1}^{j},\,j\,r_{+1}^{j},\,j^{2}\,r_{+1}^{j},\,j^{3}\,r_{+1}^{j},\qquad$
$\displaystyle S_{1}=S_{2},\,S_{1}^{2}=4$ (5atbn)

as additional solutions to (4).

######  Proof.

We focus first on $S_{1}=S_{2},\,S_{1}^{2}\neq 4$, where (5x) implies $r_{\pm
1}=r_{\pm 2}$. Substituting $\xi_{j}\propto j\,r_{\pm 1}^{j}$ into (4) and
dividing by $r_{\pm 1}^{j-2}\neq 0$ gives

$\displaystyle\qquad 0=j\left[r_{\pm 1}^{4}-\zeta r_{\pm
1}^{2}+1-\eta\left(r_{\pm 1}^{3}+r_{\pm 1}\right)\right]\,+\,2\,r_{\pm
1}^{4}-2-\eta\,(r_{\pm 1}^{3}-r_{\pm 1}).$ (5atbo)

Here, the first bracket vanishes since $r_{\pm 1}$ satisfies (4). Due to (5w),
$S_{1}=S_{2}$ implies $\eta=2S_{1}$. Using $S_{1}=r_{+1}+r_{-1}$, one has

$\displaystyle\qquad 0=2\,r_{\pm 1}^{4}-2-\eta\,(r_{\pm 1}^{3}-r_{\pm
1})=2\,r_{\pm 1}^{4}-2-2(r_{+1}+r_{-1})\,(r_{\pm 1}^{3}-r_{\pm 1}).$ (5atbp)

Recalling $r_{+1}r_{-1}=1$ and (5atbo) is satisfied indeed. Thus,
$\xi_{j}\propto j\,r_{\pm 1}^{j}$ is a symmetric Tetranacci polynomial.

For $S_{1}=S_{2},\,S_{1}^{2}=4$, (5x), states $r_{+1}=r_{-1}=r_{+2}=r_{-2}$.
Due to $S_{1}=S_{2}$, $jr_{+1}^{j}$ is already a solution and we turn directly
to $j^{2}r_{1}^{j}$. From (4), we find

$\displaystyle\qquad 0$ $\displaystyle=j^{2}\left[r_{\pm 1}^{4}-\zeta r_{\pm
1}^{2}+1-\eta\left(r_{\pm 1}^{3}+r_{\pm 1}\right)\right]\,+\,2j\left[2\,r_{\pm
1}^{4}-2-\eta\,(r_{\pm 1}^{3}-r_{\pm 1})\right],$
$\displaystyle\qquad+4\,r_{\pm 1}^{4}+4-\eta\,(r_{\pm 1}^{3}+r_{\pm 1}).$
(5atbq)

Here, the first line vanishes. Since (5x) implies $r_{+1}=S_{1}/2$ at
$S_{1}=S_{2}$, $S_{1}^{2}=4$, we find

$\displaystyle\qquad 4\,r_{\pm 1}^{4}+4-\eta\,(r_{\pm 1}^{3}+r_{\pm
1})\,=\,4\left(\frac{S_{1}}{2}\right)^{4}-4\frac{S_{1}}{2}\left(\frac{S_{1}}{2}\right)^{3}+4-S_{1}^{2}=0.$
(5atbr)

Thus, $j^{2}\,r_{+1}^{j}$ is a solution to (4). Further, $j^{3}\,r_{+1}^{j}$
also satisfies (4). Similar as before, terms associated to $j^{2},j,j^{0}$
drop. We arrive at

$\displaystyle\qquad 0\,=\,8r_{+1}^{4}-8-\eta\,r_{+1}^{3}+\eta\,r_{+1}$
$\displaystyle=8\left(\frac{S_{1}}{2}\right)^{4}-8-\eta\,r_{+1}\left(r_{+1}^{2}-1\right)$
(5atbs)
$\displaystyle=8\left(\frac{S_{1}}{2}\right)^{4}-8-\eta\,r_{+1}\left(\frac{S_{1}^{2}}{4}-1\right)\equiv
0,$

due to $r_{+1}=S_{1}/2$, $S_{1}^{2}=4$. Hence, $j^{3}\,r_{+1}^{j}$ satisfies
(4). $\square$

##  Appendix B $\mathcal{T}_{-1}(j)$, $\mathcal{T}_{0}(j)$,
$\mathcal{T}_{1}(j)$ for degenerate roots

For $S_{1}=S_{2}\neq\pm 2$, we have

$\displaystyle\qquad\mathcal{T}_{-1}(j)$
$\displaystyle=\frac{3j\varphi_{1}(j+2)-(j+2)(S_{1}^{2}-1)\varphi_{1}(j)}{S_{1}^{2}-4},$
(5atbt) $\displaystyle\qquad\mathcal{T}_{0}(j)$
$\displaystyle=\frac{2(S_{1}^{2}-1)(j+2)\,\varphi_{1}(j+1)-3(j+1)S_{1}\,\varphi_{1}(j+2)}{S_{1}^{2}-4},$
(5atbu) $\displaystyle\qquad\mathcal{T}_{1}(j)$
$\displaystyle=\frac{j\varphi_{1}(j+2)-(j+2)\varphi_{1}(j)}{S_{1}^{2}-4},$
(5atbv)

while $S_{1}=S_{2}=\pm 2$ gives

$\displaystyle\qquad\mathcal{T}_{-1}(j)$
$\displaystyle=S_{1}\frac{(2-j)j\,\varphi_{1}(j-1)+2S_{1}(j^{2}-1)\,\varphi_{1}(j)}{12},$
(5atbw) $\displaystyle\qquad\mathcal{T}_{0}(j)$
$\displaystyle=S_{1}\frac{(3+j)(1+j)\,\varphi_{1}(j+2)-2S_{1}(j+2)j\,\varphi_{1}(j+1)}{12},$
(5atbx) $\displaystyle\qquad\mathcal{T}_{1}(j)$
$\displaystyle=S_{1}\frac{(2+j)j\,\varphi_{1}(j+1)}{12}$ (5atby)

for generic integer $j$.

######  Proof.

The displayed formulae follow directly by substituting (5aj) into the
relations from Lemmata 2.1, 2.2 and exploiting the properties of
$\varphi_{1,2}(j)$ drawn in Proposition 2 and Theorem 3.1. $\square$

##  Appendix C Tetranacci polynomials in quantum transport

When applicable, Tetranacci polynomials own the potential to provide exact
analytic results for the linear and non-linear transport of quantum and
mesoscopic devices. However, the unknown non-linear identities between
Tetranacci polynomials (if existent) impose serious challenges in simplifying
the retarded Green’s function (see below). Captured by the explicit structure
of the coefficients $\zeta$, $\eta$, model specific properties may still allow
substantial progress as has been shown in case of the Kitaev chain [21, 31].
Unfortunately, this seems not to be true for the atomic chain discussed in
section 4. However, the system actually allows for a short and evident
connection between Tetranacci polynomials, the retarded Green’s function
$G^{r}$ and the transmission probability $\mathcal{T}(E)$. We provide a sketch
on the basic strategy.

We consider two normal conducting, non interacting contacts ($\alpha=L,R$)

$\displaystyle\hat{H}_{\alpha}=\sum\limits_{k}\epsilon_{k\alpha}\,c_{k\alpha}^{\dagger}c_{k\alpha}$
(5atbz)

sandwiching the atomic chain $\hat{H}$ stated in (5aq). Here,
$\epsilon_{k\alpha}^{(\dagger)}$ removes (adds) a spinless electron from (to)
contact $\alpha$ and $\epsilon_{k\alpha}$ denotes the electron’s energy. The
tunneling Hamiltonian

$\displaystyle\hat{H}_{\rm
T}=\sum\limits_{k}\left(t_{L}(k)\,d_{1}^{\dagger}c_{kL}\,+\,t_{R}(k)\,d_{N}^{\dagger}c_{kR}\right)\,+\,{\rm
h.c.}$ (5atca)

allows the exchange of particles only close to the chain’s ends for
simplicity. An applied bias $eV$ between the two contacts initializes an
electric current $I(t)=-e\,d\langle\hat{N}_{\alpha}\rangle/(dt)$, $e>0$. The
Hamiltonian $\hat{H}_{\rm tot}=\hat{H}+\hat{H}_{\rm
T}+\hat{H}_{L}+\hat{H}_{R}$ captures the time evolution of the particle number
operator $\hat{N}_{\alpha}=\sum_{k}c_{k\alpha}^{\dagger}c_{k\alpha}$. Applying
the non-equilibrium Green’s function technique yields straightforwardly the
steady state current [30, 46, 47, 48]

$\displaystyle
I=\frac{e}{h}\int\limits_{-\infty}^{\infty}dE~{}\mathcal{T}(E)\left[f(E)-f(E+eV)\right],$
(5atcb)

after some algebra. Here, $f(E)$ abbreviates the Fermi function and $h$ is
Planck’s constant. The self-energies $\Sigma^{r}_{L,R}$ ($i,j=1,\,\ldots,\,N$)

$\displaystyle(\Sigma^{r}_{L})_{i,j}$
$\displaystyle=(\Lambda_{L}-\mathrm{i}\gamma_{L})\,\delta_{1i}\delta_{1j},$
(5atcca) $\displaystyle(\Sigma^{r}_{R})_{i,j}$
$\displaystyle=(\Lambda_{R}-\mathrm{i}\gamma_{R})\,\delta_{Ni}\delta_{Nj}$
(5atccb)

account for the coupling of leads and chain. They are sparse matrices due to
the choice of $\hat{H}_{\rm T}$. The quantities
$\gamma_{\alpha}=\pi\sum_{k}|t_{\alpha}(k)|^{2}\,\delta(E-\epsilon_{k\alpha})$,
$\Lambda_{\alpha}=\mathcal{P}\sum_{k}|t_{\alpha}(k)|^{2}/(E-\epsilon_{k\alpha})$
are real and $\mathcal{P}$ denotes Cauchy’s principle value. The transmission
probability $\mathcal{T}(E)={\rm Tr}\\{\Gamma_{L}G^{r}\Gamma_{R}G^{a}\\}$ is
given as a trace over broadening matrices
$\Gamma_{\alpha}=-2\,\mathrm{Im}(\Sigma^{r}_{\alpha})$ and retarded (advanced)
Green’s functions $G^{r}$ ($G^{a}$). Due to $G^{a}=(G^{r})^{\dagger}$,
$G^{r}=(E\mathds{1}_{N}-\mathcal{H}-\Sigma_{L}^{r}-\Sigma_{R}^{r})^{-1}$ and
$\mathcal{H}$ from (5ar), $\mathcal{T}(E)$ is fully determined.

Contrary to the full non-linear transport regime in (5atcb), the linear
conductance

$\displaystyle G=\lim\limits_{eV\rightarrow 0}\frac{\partial I}{\partial
V}=\frac{e^{2}}{h}\mathcal{T}(E=0).$ (5atcccd)

relates to $\mathcal{T}(0)$ only for zero temperature. The sparsity of
$\Gamma_{L,R}$ yields generally
$\mathcal{T}(E)=4\gamma_{L}\gamma_{R}|G^{r}_{1N}|^{2}$. Instead of performing
the inversion, $G^{r}_{1N}$ can be obtained much easier using Tetranacci
poylnomials. Since the connection is more apparent for $E\neq 0$, we focus on
$\mathcal{T}(E)$ instead.

Obviously, $G^{r}$ obeys
$(E\mathds{1}_{N}-\mathcal{H}-\Sigma_{L}^{r}-\Sigma_{R}^{r})G^{r}=\mathds{1}_{N}$.
As we shall see, we can approach as for the eigenvector equation of
$\mathcal{H}$. However, notice that $E$ is a continuous variable and not
necessarily an eigenvalue of $\mathcal{H}$. When
$G^{r}=(\vec{v}_{1},\,\ldots,\,\vec{v}_{N})^{\mathrm{T}}$ is written in terms
of column vectors $\vec{v}_{i}\in\mathbb{R}^{N\times 1}$, those obey
($i=1\,\ldots,\,N$)

$\displaystyle(E\mathds{1}_{N}-\mathcal{H}-\Sigma_{L}^{r}-\Sigma_{R}^{r})\vec{v}_{i}=\vec{e}_{i}$
(5atccce)

where $\vec{e}_{i}$ abbreviates the $i$-th column of $\mathds{1}_{N}$. The
structure of (5atccce) reminds an eigenvector equation for the matrix
$\mathcal{H}+\Sigma_{L}^{r}+\Sigma_{R}^{r}$, but the inhomoginity
$\vec{e}_{i}$ yields a unique solution for $\vec{v}_{i}$. Aiming on
$G^{r}_{1N}$, sets $i=N$ in (5atccce). Defining
$\vec{v}_{N}=(\sigma_{1},\ldots,\,\sigma_{N})^{\mathrm{T}}$ grants
$\sigma_{j+2}=\zeta\,\xi_{j}-\xi_{j-2}+\eta\,(\xi_{j+1}+\xi_{j-1})$ with
$\zeta=-(E+\mu)/t_{2}$, $\eta=-t_{1}/t_{2}$ for $j=3\,\ldots,N-3$ similar to
the eigenvector equation of $\mathcal{H}$. Yet, the self-energies modify the
boundary conditions. Extending again the recursion to all integers $j$, but
keeping $\vec{v}_{N}$ untouched, the constraints reduce to

$\displaystyle 0=$ $\displaystyle\sigma_{0}$ (5atcccfa) $\displaystyle 0=$
$\displaystyle\sigma_{N+1},$ (5atcccfb) $\displaystyle 0=$
$\displaystyle(\mathrm{i}\gamma_{L}-\Lambda_{L})\sigma_{1}-t_{2}\sigma_{-1},$
(5atcccfc) $\displaystyle 1=$
$\displaystyle(\mathrm{i}\gamma_{R}-\Lambda_{R})\sigma_{N}-t_{2}\sigma_{N+2},$
(5atcccfd)

for $t_{2}\neq 0$. Here ”$1$” originates from $(\vec{v}_{N})_{N}=1$. Due to
Corollary 1, we search for $g_{-2}\,\ldots,\,g_{-1}$. From
(5atcccfa)-(5atcccfd), we find

$\displaystyle\sigma_{j}=g_{-2}\mathcal{T}_{-2}(j)+g_{1}\left[\mathcal{T}_{1}(j)+\frac{\mathrm{i}\gamma_{L}-\Lambda_{L}}{t_{2}}\mathcal{T}_{-1}(j)\right]$
(5atcccfcg)

thus a $2\times 2$ matrix equation

$\displaystyle\left(\matrix{a&b\cr c&d}ß\right)\left(\matrix{g_{-2}\cr
g_{1}}ß\right)=\left(\matrix{0\cr 1}ß\right)$ (5atcccfch)

with coefficients

$\displaystyle a$ $\displaystyle=\mathcal{T}_{-2}(N+1),$ (5atcccfcia)
$\displaystyle b$
$\displaystyle=\mathcal{T}_{1}(N+1)+\frac{\mathrm{i}\gamma_{L}-\Lambda_{L}}{t_{2}}\mathcal{T}_{-1}(N+1),$
(5atcccfcib) $\displaystyle c$
$\displaystyle=(\mathrm{i}\gamma_{R}-\Lambda_{R})\mathcal{T}_{-2}(N)-t_{2}\mathcal{T}_{1}(N+2),$
(5atcccfcic) $\displaystyle d$
$\displaystyle=(\mathrm{i}\gamma_{R}-\Lambda_{R})\left[\mathcal{T}_{1}(N)+\frac{\mathrm{i}\gamma_{L}-\Lambda_{L}}{t_{2}}\,\mathcal{T}_{-1}(N)\right]$
(5atcccfcid) $\displaystyle\quad-
t_{2}\left[\mathcal{T}_{1}(N+2)+\frac{\mathrm{i}\gamma_{L}-\Lambda_{L}}{t_{2}}\,\mathcal{T}_{-1}(N+2)\right].$

The solution to (5atcccfch) is

$\displaystyle\left(\matrix{g_{-2}\cr g_{1}}ß\right)=\frac{1}{ab-
cd}\left(\matrix{-\mathcal{T}_{1}(N+1)-\frac{\mathrm{i}\gamma_{L}-\Lambda_{L}}{t_{2}}\mathcal{T}_{-1}(N+1)\cr\mathcal{T}_{-2}(N+1)}ß\right).$
(5atcccfcicj)

Hence, $G^{r}_{1N}\equiv\sigma_{N}$ contains non-linear combinations of
$\mathcal{T}_{i}(N),\,\mathcal{T}_{i}(N+1)$ ($i=-2,\,\ldots,1$) when
(5atcccfcicj) is inserted into (5atcccfcg).

For zero temperature, the linear conductance depends on
$\left.G^{r}_{1N}\right|_{E=0}$ and non-linear identities can possibly be
circumvented. In case of the Kitaev chain, the setting $E=0$ turns all
Tetranacci into Fibonacci polynomials [21]. However, this is a model specific
property imprinted into the respective composition of $\zeta$, $\eta$ in terms
of the model’s parameters. Unfortunately, the nearest neighbor hopping chain
misses this feature and further progress towards an useful explicit formula
for $\left.G^{r}_{1N}\right|_{E=0}$, $\mathcal{T}(0)$ is prohibited.

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