# Relations between Chebyshev, Fibonacci and Lucas polynomials via
trigonometric sums

Lejla Smajlović  School of Economics and Business
University of Sarajevo
Trg Oslobodjenja Alija Izetbegović 1
71 000 Sarajevo
Bosnia and Herzegovina<EMAIL_ADDRESS>,  Zenan Šabanac
Department of Mathematics and Computer Science
University of Sarajevo
Zmaja od Bosne 33-35
71 000 Sarajevo
Bosnia and Herzegovina<EMAIL_ADDRESS>and  Lamija Šćeta  School of
Economics and Business
University of Sarajevo
Trg Oslobodjenja Alija Izetbegović 1
71 000 Sarajevo
Bosnia and Herzegovina<EMAIL_ADDRESS>
###### Abstract.

In this paper we derive some new identities involving the Fibonacci and Lucas
polynomials and the Chebyshev polynomials of the first and the second kind.
Our starting point is a finite trigonometric sum which equals the resolvent
kernel on the discrete circle with $m$ vertices and which can be evaluated in
two different ways. An expression for this sum in terms of the Chebyshev
polynomials was deduced in [18] and the expression in terms of the Fibonacci
and Lucas polynomials is deduced in this paper. As a consequence, we establish
some further identities involving trigonometric sums and Fibonacci, Lucas,
Pell and Pell-Lucas polynomials and numbers, thus providing a ”physical”
interpretation for those identities. Moreover, the finite trigonometric sum of
the type considered in this paper can be related to the effective resistance
between any two vertices of the $N$-cycle graph with four nearest neighbors
$C_{N}(1,2)$. This yields further identities involving Fibonacci numbers.

###### Key words and phrases:

Fibonacci and Lucas polynomials and numbers, trigonometric sums, Chebyshev
polynomials, resolvent kernel, effective resistance

###### 1991 Mathematics Subject Classification:

11B39, 33C45

##  1\. Introduction and statement of main results

The Fibonacci sequence $\left\\{F_{n}\right\\}_{n=0}^{\infty}$ is defined by
recursive relation

$F_{0}=0,\text{ }F_{1}=1,\text{ }F_{n}=F_{n-1}+F_{n-2}\text{ for }n\geq 2,$

and the Lucas sequence $\left\\{L_{n}\right\\}_{n=0}^{\infty}$ is defined by

$L_{0}=2,\text{ }L_{1}=1,\text{ }L_{n}=L_{n-1}+L_{n-2}\text{ for }n\geq 2.$

Both sequences arise in many applications across many mathematical and non-
mathematical disciplines, such as art, architecture, biology, chemistry,
chess, electrical engineering, music, origami, poetry, physics, physiology,
neurophysiology, and many others (see [21]). Recent applications include
algebraic coding theory, linear sequential circuits, quasicrystals,
phyllotaxies, biomathematics, and computer science (see, e.g. [12, 16, 26]).

A vast collection of interesting identities and many other properties
involving Fibonacci and Lucas numbers, as well as other integer sequences, can
be found in [21, 22]. The original proofs of many of these identities employ
diverse techniques, including reciprocal sums, Binet-like formulas, generating
functions, tiling, combinatorial approaches, determinant and matrix methods.

The Fibonacci polynomials are defined by the recurrence relation:

(1.1) $F_{0}(x)=0,\text{ }F_{1}(x)=1,\text{
}F_{n}(x)=xF_{n-1}(x)+F_{n-2}(x)\text{ for }n\geq 2\text{.}$

The Lucas polynomials satisfy the same recurrence relation, with a different
initial condition:

(1.2) $L_{0}(x)=2\text{, }L_{1}(x)=x\text{, and
}L_{n}(x)=xL_{n-1}(x)+L_{n-2}(x)\text{ for }n\geq 2\text{.}$

The Fibonacci and Lucas numbers are special values of the Fibonacci and Lucas
polynomials: $F_{n}(1)=F_{n}$ and $L_{n}(1)=L_{n}$ for nonnegative integers
$n$.

The Fibonacci and Lucas polynomials and their generalizations appear in
different applications in various disciplines such as physics, biology,
computer science, and statistics, see [21, 22]. For example, both fractional
and higher-order derivatives can be easily calculated using Lucas and
Fibonacci polynomial relations, see [9, 19]. Lucas polynomials arise in
numerical methods for solving differential and integral equations [7, 23, 25,
4, 24]. The expressions for the Fibonacci and Lucas polynomials in terms of
the Chebyshev polynomials have been derived in [3, 2, 20] and corollaries of
[1, Theorem 3], however, the coefficients are rather complicated as they are
expressed in terms of the hypergeometric functions. This is expected, because
the recurrence relations (1.1) and (1.2) indicate that the Chebyshev
polynomials of the first and the second kind are equivalent to the Fibonacci
and Lucas polynomials, when viewed as the generalized Fibonacci polynomial
sequences, see [13].

Many identities involving the Fibonacci and Lucas polynomials have been
discovered so far. A very long list of those identities is provided in [22],
see also [32, 34, 14]. The aim of this paper is to provide a ”physical”
interpretation behind the new and the known identities. Namely, the identities
derived in this paper stem from different evaluations of the same
trigonometric sum, which possesses a physical interpretation. Let us describe
our two main results in a greater detail.

The starting point for our first main result is an observation that the zeros
of Fibonacci and Lucas polynomials can be expressed in terms of basic
trigonometric functions (see formula (2.17) below). Therefore, the logarithmic
derivative of those polynomials is equal to a certain finite trigonometric
sum. Curiously, this sum, after some trivial manipulations can be identified
with the multiple of a special value of the resolvent kernel on the discrete
circle $X_{m}$ with $m$ vertices, twisted by an additive character. By using
results of [18] in which this sum is evaluated in terms of the Chebyshev
polynomials, we deduce a new identity relating the Fibonacci and Lucas
polynomials with the Chebyshev polynomials of the first and the second kind,
as stated in the following theorem.

######  Theorem 1.1.

The following identities hold true for all positive integers $n$:

(1.3)
$\frac{L_{n}\left(2x\right)}{4x\left(x^{2}+1\right)F_{n}\left(2x\right)}=\frac{U_{n-1}\left(2x^{2}+1\right)}{T_{n}\left(2x^{2}+1\right)+\left(-1\right)^{n-1}}$

and

(1.4)
$\frac{F_{n}\left(2x\right)}{xL_{n}\left(2x\right)}=\frac{U_{n-1}\left(2x^{2}+1\right)}{T_{n}\left(2x^{2}+1\right)+\left(-1\right)^{n}}.$

As a corollary, we derive identities relating certain finite trigonometric
sums and the Fibonacci, Lucas, Pell and Pell-Lucas polynomials. Moreover, by
using the relation (2.20) below between the Chebyshev and the Lucas and
Fibonacci polynomials, from Theorem 1.1 we immediately derive the following
functional relation between Chebyshev polynomials evaluated at different
arguments.

######  Corollary 1.2.

For any positive integer $n$ and all $x\in\mathbb{R}$ we have

$T_{n}(x)(1-T_{n}(2x^{2}-1))=x(x^{2}-1)U_{n-1}(x)U_{n-1}(2x^{2}-1).$

The second main result, given in Theorem 1.3 below is also deduced by starting
with a certain finite trigonometric sum which possesses a physical
interpretation; namely it can be viewed as the effective resistance between
two nodes in a certain special resistor network (see, e.g. [10, 11, 33]). This
sum is evaluated in [11] in terms of the Fibonacci numbers. A different
evaluation of this sum in terms of special values of the Chebyshev polynomials
yields the following theorem.

######  Theorem 1.3.

For a positive integer $N$, let

(1.5)
$C_{N}=\left(1+\left(\frac{3-\sqrt{5}}{2}\right)^{N}\right)\left(1-\left(\frac{3-\sqrt{5}}{2}\right)^{N}\right)^{-1}.$

The following identity holds true

(1.6)
$\frac{F_{2N-2\ell}+(-1)^{N}F_{2\ell}+\left(-1\right)^{\ell+1}F_{2N}}{\frac{1}{2}L_{2N}-(-1)^{N}}=\sqrt{5}F_{\ell}^{2}C_{N}^{(-1)^{N}}-F_{2\ell}.$

As corollaries we derive further identities involving Fibonacci and Lucas
numbers as well as Fibonacci hyperbolic sine and cosine functions.

The paper is organized as follows: In Section 2 we present some known
relationships between the Chebyshev, Fibonacci, Lucas, Pell and Pell-Lucas
polynomials and the Fibonacci, Lucas, Pell and Pell-Lucas numbers, as well as
the Fibonacci hyperbolic sine and cosine functions. In Section 3 we recall
evaluations of certain trigonometric sums in terms of the Chebyshev
polynomials and the Fibonacci and Lucas numbers, stemming from the analysis of
the resolvent kernel and the resistance distance on circular graphs. In
Section 4 we prove our two main theorems, while in Section 5 we establish some
further identities involving the Fibonacci, Lucas, Pell and Pell-Lucas
polynomials and numbers as well as the Fibonacci hyperbolic sine and cosine
functions and the Bejaia and Pisa numbers.

##  2\. Preliminaries

In this section we recall some well-known properties of the Chebyshev
polynomials of the first and the second kind, the Fibonacci and Lucas
polynomials and the Fibonacci hyperbolic sine and cosine functions.

###  2.1. Chebyshev polynomials and their relations with Fibonacci and Lucas
numbers

The Chebyshev polynomials are sequences of orthogonal polynomials related to
the cosine and sine functions. The Chebyshev polynomials of the first kind,
denoted by $T_{n}$, $n\in\mathbb{N}\cup\\{0\\}$ are the unique polynomials
satisfying

$T_{n}\left(\cos\theta\right)=\cos\left(n\theta\right),\,\,n=0,1,2,\ldots.$

As functions of a complex variable $x$ they can be defined as

$T_{n}\left(x\right)=\frac{1}{2}\left(\left(x-\sqrt{x^{2}-1}\right)^{n}+\left(x+\sqrt{x^{2}-1}\right)^{n}\right),$

where, as is standard, we assume that the branch of the square root is
principal.

The Chebyshev polynomials of the second kind, denoted by $U_{n}$,
$n\in\mathbb{N}\cup\\{0\\}$ are defined as the unique polynomials satisfying

$U_{n}\left(\cos\theta\right)\sin\theta=\sin\left(\left(n+1\right)\theta\right),\,\,n=0,1,2,\ldots.$

A very good reference, containing information about numerous applications of
the Chebyshev polynomials in approximation theory, algebra, number theory,
ergodic theory, and numerical analysis is [27].

The Chebyshev polynomials possess the following well-known symmetry properties

(2.1) $\displaystyle T_{n}\left(-x\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}T_{n}\left(x\right),$ (2.2) $\displaystyle
U_{n}\left(-x\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}U_{n}\left(x\right).$

There are various relations between the values of Chebyshev polynomials for
specific values of the variable and the Fibonacci and Lucas numbers (see e.g.
[6, Section 2], [15, Remark 9] or [35]). Below, we list some of them:

(2.3) $\displaystyle T_{n}\left(\frac{3}{2}\right)$
$\displaystyle=\frac{L_{2n}}{2};\text{ \ \
}U_{n}\left(\frac{3}{2}\right)=F_{2n+2};$ (2.4) $\displaystyle
T_{n}\left(\frac{7}{2}\right)$ $\displaystyle=\frac{L_{4n}}{2};\text{ \ \
}U_{n}\left(\frac{7}{2}\right)=\frac{F_{4n+4}}{3};$ (2.9) $\displaystyle
T_{n}\left(\frac{\sqrt{5}}{2}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{L_{n}}{2},&\text{if }n\text{
is even;}\\\ \frac{\sqrt{5}F_{n}}{2},&\text{if }n\text{ is
odd;}\end{array}\right.\text{ \ \
}U_{n}\left(\frac{\sqrt{5}}{2}\right)=\left\\{\begin{array}[]{ll}L_{n+1},&\text{if
}n\text{ is even;}\\\ \sqrt{5}F_{n+1},&\text{if }n\text{ is
odd;}\end{array}\right.\text{ }$ (2.14) $\displaystyle
T_{n}\left(\sqrt{5}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{L_{3n}}{2},&\text{if }n\text{
is even;}\\\ \frac{\sqrt{5}F_{3n}}{2},&\text{if }n\text{ is
odd;}\end{array}\right.\text{ \ \
}U_{n}\left(\sqrt{5}\right)=\left\\{\begin{array}[]{ll}\frac{L_{3n+3}}{4},&\text{if
}n\text{ is even;}\\\ \frac{\sqrt{5}F_{3n+3}}{4},&\text{if }n\text{ is
odd.}\end{array}\right.$

By substituting $m$ by $2m$ and $x=\frac{\sqrt{5}}{2}$ in the equality
$T_{n}\left(T_{m}\left(x\right)\right)=T_{nm}\left(x\right)$, and using above
identities, we obtain

(2.15) $T_{n}\left(\frac{1}{2}L_{2m}\right)=\frac{1}{2}L_{2nm}.$

For the Chebyshev polynomials of the second kind we have (see [6, Section 2])

(2.16) $U_{n-1}\left(\frac{1}{2}L_{2m}\right)=\frac{F_{2nm}}{F_{2m}}.$

###  2.2. Fibonacci and Lucas polynomials

The Fibonacci and Lucas polynomials are defined by the recurrence relations
(1.1) and (1.2). Hoggatt and Bicknell [17] proved that the roots of the
Fibonacci and Lucas polynomials are given as follows:

(2.17) $\begin{array}[]{lcll}F_{2n}\left(x\right)=0\text{ }&\text{at }&x=\pm
2i\sin\frac{k\pi}{2n},&k=0,1,\ldots,n-1,\\\ F_{2n+1}\left(x\right)=0\text{
}&\text{at }&x=\pm
2i\sin\frac{\left(k+1/2\right)\pi}{2n+1},&k=0,1,\ldots,n-1,\\\
L_{2n}\left(x\right)=0\text{ }&\text{at }&x=\pm
2i\sin\frac{\left(k+1/2\right)\pi}{2n},&k=0,1,\ldots,n-1,\\\
L_{2n+1}\left(x\right)=0\text{ }&\text{at }&x=\pm
2i\sin\frac{k\pi}{2n+1},&k=0,1,\ldots,n.\end{array}$

The first derivative of the Lucas polynomial can be expressed as

(2.18) $L_{n}^{\prime}\left(x\right)=nF_{n}\left(x\right),$

see [22, formula (31.22) on p. 15]). The first derivative of the Fibonacci
polynomial can easily be derived using the formula at the bottom of p. 15 of
[22] to get

(2.19)
$F_{n}^{\prime}\left(x\right)=\frac{nL_{n}\left(x\right)-xF_{n}\left(x\right)}{x^{2}+4}.$

An in-depth study of the Fibonacci and Lucas polynomials can be found in [22,
Chapters 31 and 32]. Moreover, in [22, Chapter 41], it is proved that for
$n\geq 1$ we have the following identities, relating the Lucas and Fibonacci
polynomials to the Chebyshev polynomials of the first and the second kind:

(2.20) $L_{n}(2x)=2(-i)^{n}T_{n}(ix)\quad\text{and}\quad
F_{n}(2x)=-(-i)^{n-1}U_{n-1}(ix).$

Using these polynomials, two additional classes of polynomials, the Pell
polynomials $P_{n}\left(x\right)$ and the Pell-Lucas polynomials
$Q_{n}\left(x\right)$ are defined:

(2.21) $P_{n}\left(x\right)=F_{n}(2x)=-(-i)^{n-1}U_{n-1}(ix)\text{ and
}Q_{n}\left(x\right)=L_{n}(2x)=2(-i)^{n}T_{n}(ix).$

The corresponding Pell numbers $P_{n}$ and the Pell-Lucas numbers $Q_{n}$ are
given by:

$P_{n}=P_{n}(1)=F_{n}(2)=-(-i)^{n-1}U_{n-1}(i)$

and

$Q_{n}=\frac{1}{2}Q_{n}(1)=\frac{1}{2}L_{n}(2)=(-i)^{n}T_{n}(i).$

For further details and properties of Pell and Pell-Lucas polynomials and
numbers, we refer an interested reader to [22, Section 31.6].

The values of Chebyshev polynomials at $x=3$ are also related to the Pell and
Pell-Lucas numbers by (see [22, Exercises 65 and 66 on p. 589]):

(2.22) $U_{n-1}\left(3\right)=\frac{1}{2}P_{2n}\text{ and
}T_{n}\left(3\right)=Q_{2n}.$

###  2.3. Fibonacci hyperbolic sine and cosine functions

For $x\in(-\infty,+\infty)$, Stakhov and Tkachenko [30] (see also [31, 29])
introduced the Fibonacci hyperbolic sine and cosine functions by analogy to
the classic hyperbolic functions:

$\mathrm{sFh}\left(x\right)=\frac{\psi^{x}-\psi^{-x}}{\sqrt{5}}\quad\text{ and
}\quad\mathrm{cFh}\left(x\right)=\frac{\psi^{\left(x+\frac{1}{2}\right)}+\psi^{-\left(x+\frac{1}{2}\right)}}{\sqrt{5}},$

where $\psi=1+\phi=\frac{3+\sqrt{5}}{2}$, and $\phi=\frac{1+\sqrt{5}}{2}$
denotes the golden ratio. Using the well-known identity $\phi^{2}=1+\phi$, one
can obtain

$\mathrm{sFh}\left(x\right)=\frac{\phi^{2x}-\phi^{-2x}}{\sqrt{5}}\quad\text{
and
}\quad\mathrm{cFh}\left(x\right)=\frac{\phi^{\left(2x+1\right)}+\phi^{-\left(2x+1\right)}}{\sqrt{5}}.$

For $k\in\mathbb{Z}$, the Fibonacci hyperbolic sine and cosine functions are
related to the classical hyperbolic sine and cosine functions by (see [31,
relation (12)]):

(2.23)
$\mathrm{sFh}\left(k\right)=\frac{2}{\sqrt{5}}\sinh\left(2k\log\phi\right)\text{
and
}\mathrm{cFh}\left(k\right)=\frac{2}{\sqrt{5}}\cosh\left(\left(2k+1\right)\log\phi\right).$

For nonnegative integers $k$, functions $\mathrm{sFh}\left(k\right)$ and
$\mathrm{cFh}\left(k\right)$ can be expressed in terms of the corresponding
elements of the Fibonacci sequence (see [31, relation (6)]):

(2.24) $\mathrm{sFh}\left(k\right)=F_{2k}\quad\text{ and
}\quad\mathrm{cFh}\left(k\right)=F_{2k+1}\text{.}$

##  3\. Closed evaluation of some finite trigonometric sums

In this section we recall results of [18] and [11] in which certain finite
trigonometric sums are expressed as rational functions of Chebyshev
polynomials and in terms of Fibonacci numbers.

###  3.1. The resolvent kernel on the discrete circle

Jorgenson, Karlsson and Smajlović in [18] computed the resolvent kernel
$G_{X_{m},\chi_{\beta}}(x,y;s)$ on the discrete circle $X_{m}$ with $m$
vertices using two different methods. The first employs the spectral expansion
of the combinatorial Laplacian which acts on the space of functions on $X_{m}$
that are twisted by a character $\chi_{\beta}$, resulting in its expression as
a finite trigonometric sum, while the second one utilizes the perspective of
$X_{m}$ as a quotient space of $\mathbb{Z}$, thus expressing the resolvent
kernel as a rational function involving Chebyshev polynomials. It has been
shown that (see [18, relations (41) and (42)])

(3.1) $\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\pi
i\frac{j\ell}{m}}}{s+2\sin^{2}\left(\pi\frac{j+\beta}{m}\right)}=e^{2\pi
i\frac{\beta\ell}{m}}\cdot\frac{U_{m-\ell-1}\left(s+1\right)+e^{2\pi
i\beta}U_{\ell-1}\left(s+1\right)}{T_{m}\left(s+1\right)-\cos 2\pi\beta},$

for $\ell\in\left\\{0,1,\ldots,m-1\right\\}$ and all complex $s$. Setting
$\beta=0$ in (3.1), we derive that

(3.2) $\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\pi
i\frac{j\ell}{m}}}{s+2\sin^{2}\frac{\pi
j}{m}}=\frac{U_{m-\ell-1}\left(s+1\right)+U_{\ell-1}\left(s+1\right)}{T_{m}\left(s+1\right)-1}.$

###  3.2. Effective resistance

Computation of resistance in electric circuits, specifically the effective
resistance between two nodes in a resistor network, can be carried out using a
variety of methods. Traditional analytic methods, such as Kirchhoff’s laws,
are theoretically applicable to any network but become impractical for larger
networks (with more nodes) due to the rapidly increasing number of equations
[11]. For this reason, computations of resistance in infinite networks have
been conducted using the Green’s function [8].

In 2004 Wu [33] introduced an approach to calculate the two-point resistance
(effective resistance) in a finite resistor network in terms of the
eigenvalues and eigenvectors of the Laplacian matrix, and derived explicit
formulas for two-point resistances in regular lattices of one, two, and three
dimensions under various boundary conditions. The following summation identity
was deduced in [33, formula (62)]

(3.3)
$\frac{1}{m}\sum_{j=0}^{m-1}\frac{\cos\left(2\ell\frac{j\pi}{m}\right)}{\cosh\lambda-\cos\left(2\frac{j\pi}{m}\right)}=\frac{\cosh\left(\frac{m}{2}-\ell\right)\lambda}{\sinh\lambda\sinh\frac{m\lambda}{2}},$

for $\lambda$ real, $m\in\mathbb{N}$ and
$\ell\in\left\\{0,1,\ldots,m-1\right\\}$.

Chair [11, relations (4), (5), (19), and (22)] calculated the exact expression
for the effective resistance between any two vertices of the $N$-cycle graph
with four nearest neighbors $C_{N}(1,2)$ in terms of the effective resistance
of the $N$-cycle graph $C_{N}$, the square of the Fibonacci numbers, and the
bisected Fibonacci numbers:

(3.4) $\displaystyle R\left(\ell\right)$ $\displaystyle=$
$\displaystyle\frac{1}{5}\ell\left(1-\frac{\ell}{N}\right)+\frac{4}{5N}\sum\limits_{j=1}^{N-1}\frac{\sin^{2}\frac{j\ell\pi}{N}}{5\left(1-\frac{4}{5}\sin^{2}\frac{j\pi}{N}\right)},$
(3.5) $\displaystyle R\left(\ell\right)$ $\displaystyle=$
$\displaystyle\frac{1}{5}\ell\left(1-\frac{\ell}{N}\right)+(-1)^{\ell+1}\frac{F_{\ell}^{2}}{\sqrt{5}}C_{N}^{(-1)^{N}}+(-1)^{\ell}\frac{F_{2\ell}}{5},$

where $C_{N}$ is given by (1.5).

##  4\. Proofs of main results

In this section, we prove Theorems 1.1 and 1.3.

###  4.1. Proof of the Theorem 1.1

First we will prove (1.3). If $n=2m$ is even, using (2.17), we write $F_{2m}$
as the product

$F_{2m}\left(x\right)=x\prod\limits_{j=1}^{m-1}\left(x^{2}+4\sin^{2}\frac{j\pi}{2m}\right),$

which, after taking the logarithmic derivative yields to

$\frac{F_{2m}^{\prime}\left(x\right)}{xF_{2m}\left(x\right)}=\sum\limits_{j=1}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}+\frac{1}{x^{2}}.$

Using the formula (2.19) for the derivative of the Fibonacci polynomial, we
get

(4.1)
$\frac{2mL_{2m}\left(x\right)-xF_{2m}\left(x\right)}{x\left(x^{2}+4\right)F_{2m}\left(x\right)}=\sum\limits_{j=1}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}+\frac{1}{x^{2}}.$

By a simple change of variables, it is immediate to deduce that

$\sum_{j=1}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}=\sum_{j=m+1}^{2m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}},$

hence,

$\frac{1}{2}\sum_{j=0}^{2m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}=\sum_{j=1}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}+\frac{1}{x^{2}}+\frac{1}{x^{2}+4}.$

Combining the last formula and (4.1), we derive

(4.2)
$\displaystyle\frac{1}{2}\sum_{j=0}^{2m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=\frac{2mL_{2m}\left(x\right)-xF_{2m}\left(x\right)}{x\left(x^{2}+4\right)F_{2m}\left(x\right)}+\frac{1}{x^{2}+4}$
$\displaystyle=\frac{2mL_{2m}\left(x\right)}{x\left(x^{2}+4\right)F_{2m}\left(x\right)}.$

Setting $s=\frac{x^{2}}{2}$, $\ell=0$ and substituting $m$ by $2m$ in (3.2),
we obtain

(4.3) $\frac{1}{2}\sum_{j=0}^{2m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\pi
j}{2m}}=\frac{mU_{2m-1}\left(\frac{x^{2}}{2}+1\right)}{T_{2m}\left(\frac{x^{2}}{2}+1\right)-1}.$

From (4.2) and (4.3) it follows that

$\frac{2L_{2m}\left(x\right)}{x\left(x^{2}+4\right)F_{2m}\left(x\right)}=\frac{U_{2m-1}\left(\frac{x^{2}}{2}+1\right)}{T_{2m}\left(\frac{x^{2}}{2}+1\right)-1}.$

Replacing $x$ by $2x$ in the last formula proves (1.3) for even $n$.

For odd numbers $n=2m+1$, using (2.17), we can write $F_{n}(x)$ as the
following product

$F_{2m+1}\left(x\right)=\prod\limits_{j=0}^{m-1}\left(x^{2}+4\sin^{2}\frac{\left(j+1/2\right)\pi}{2m+1}\right).$

Upon taking the logarithmic derivative, it is straightforward to get

$\frac{\left(2m+1\right)L_{2m+1}\left(x\right)-xF_{2m+1}\left(x\right)}{x\left(x^{2}+4\right)F_{2m+1}\left(x\right)}=\frac{1}{2}\sum\limits_{j=0}^{2m}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m+1}}-\frac{1}{x^{2}+4},$

or, equivalently

$\frac{\left(2m+1\right)L_{2m+1}\left(x\right)}{x\left(x^{2}+4\right)F_{2m+1}\left(x\right)}=\frac{1}{2}\sum\limits_{j=0}^{2m}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m+1}}.$

On the other hand, using (3.1) with $s=\frac{x^{2}}{2}$, $\ell=0$, and
$\beta=\frac{1}{2}$, the right-hand side of the above display can be written
as

$\sum\limits_{j=0}^{2m}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m+1}}=\frac{\left(2m+1\right)U_{2m}\left(\frac{x^{2}}{2}+1\right)}{T_{2m+1}\left(\frac{x^{2}}{2}+1\right)+1}.$

Combining the last two equations we get

$\frac{2L_{2m+1}\left(x\right)}{x\left(x^{2}+4\right)F_{2m+1}\left(x\right)}=\frac{U_{2m}\left(\frac{x^{2}}{2}+1\right)}{T_{2m+1}\left(\frac{x^{2}}{2}+1\right)+1}.$

Substituting $x$ by $2x$ proves (1.3) for $n$ odd.

Now we prove (1.4). For $n=2m$ we have

$L_{2m}\left(x\right)=\prod\limits_{j=0}^{m-1}\left(x^{2}+4\sin^{2}\frac{\left(j+1/2\right)\pi}{2m}\right).$

Taking the logarithmic derivative of the above display, and using (2.18) we
obtain

$\frac{2mF_{2m}\left(x\right)}{xL_{2m}\left(x\right)}=\sum\limits_{j=0}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m}}.$

Using the symmetry property

$\sum\limits_{j=0}^{m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m}}=\sum\limits_{j=m}^{2m-1}\frac{1}{\frac{x^{2}}{2}+2\sin^{2}\frac{\left(j+1/2\right)\pi}{2m}},$

and substituting $m$ by $2m$ in (3.1) with $s=\frac{x^{2}}{2}$, $\ell=0$, and
$\beta=\frac{1}{2}$, we derive that

$\frac{2F_{2m}\left(x\right)}{xL_{2m}\left(x\right)}=\frac{U_{2m-1}\left(\frac{x^{2}}{2}+1\right)}{T_{2m}\left(\frac{x^{2}}{2}+1\right)+1}.$

Replacing $x$ by $2x$ in the last formula proves (1.4) in case of $n$ even.
The proof of (1.4) for odd $n$ is analogous, hence we omit it here.

###  4.2. Proof of the Theorem 1.3

Comparing relations (3.4) and (3.5), we deduce that

(4.4)
$\frac{4}{5N}\sum\limits_{j=1}^{N-1}\frac{\sin^{2}\frac{j\ell\pi}{N}}{1-\frac{4}{5}\sin^{2}\frac{j\pi}{N}}=\left(-1\right)^{\ell+1}\sqrt{5}F_{\ell}^{2}C_{N}^{(-1)^{N}}+\left(-1\right)^{\ell}F_{2\ell},$

where $C_{N}$ is given by (1.5). On the other hand, we have:

$\displaystyle\frac{4}{5N}\sum\limits_{j=1}^{N-1}\frac{\sin^{2}\frac{j\ell\pi}{N}}{1-\frac{4}{5}\sin^{2}\frac{j\pi}{N}}=$
$\displaystyle-\frac{1}{N}\sum\limits_{j=0}^{N-1}\frac{e^{\frac{2j\pi
i}{N}\cdot
0}}{-\frac{5}{2}+2\sin^{2}\frac{j\pi}{N}}+\frac{1}{2N}\sum\limits_{j=0}^{N-1}\frac{e^{\frac{2j\ell\pi}{N}i}}{-\frac{5}{2}+2\sin^{2}\frac{j\pi}{N}}$
$\displaystyle+\frac{1}{2N}\sum\limits_{j=0}^{N-1}\frac{e^{\frac{2j\left(N-\ell\right)\pi
i}{N}}}{-\frac{5}{2}+2\sin^{2}\frac{j\pi}{N}}.$

Using (3.2) with $s=-\frac{5}{2}$, and fact that $U_{-1}\left(x\right)=0$ for
all $x$, we obtain

(4.5)
$\frac{4}{5N}\sum\limits_{j=1}^{N-1}\frac{\sin^{2}\frac{j\ell\pi}{N}}{1-\frac{4}{5}\sin^{2}\frac{j\pi}{N}}=\frac{U_{N-\ell-1}\left(-\frac{3}{2}\right)+U_{\ell-1}\left(-\frac{3}{2}\right)-U_{N-1}\left(-\frac{3}{2}\right)}{T_{N}\left(-\frac{3}{2}\right)-1}.$

Applying the symmetry properties (2.1) and (2.2) and relation (2.3), we obtain

$T_{n}\left(-\frac{3}{2}\right)=\left(-1\right)^{n}\frac{L_{2n}}{2},\text{ \ \
}U_{n}\left(-\frac{3}{2}\right)=\left(-1\right)^{n}F_{2n+2}.$

Combining the above equation with (4.5) and (4.4) yields (1.6).

##  5\. Further identities

In this section we derive many identities stemming from deeper analysis of the
relation between finite trigonometric sums considered in this paper and
certain polynomials and integer sequences satisfying the second order
recursion relations.

###  5.1. Identities relating trigonometric sums and Fibonacci, Lucas, Pell
and Pell-Lucas polynomials

Using the relations which arose in the proof of Theorem 1.1, one can derive
further identities, as listed in the following corollary.

######  Corollary 5.1.

For any positive real number $x$, and any positive integer $m$ the following
formulas hold true:

(5.1) $\displaystyle\sum_{j=0}^{2m-1}\frac{1}{x^{2}+\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=\frac{\left(2m-1\right)F_{2m+1}\left(2x\right)+\left(2m+1\right)F_{2m-1}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}+\frac{1}{x^{2}+1},$
(5.2) $\displaystyle\sum_{j=0}^{2m-1}\frac{1}{x^{2}+4\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=\frac{2mL_{2m}\left(x\right)-xF_{2m}\left(x\right)}{x\left(x^{2}+4\right)F_{2m}\left(x\right)}+\frac{1}{x^{2}+4}.$
(5.3)
$\sum_{j=0}^{2m-1}\frac{1}{x^{2}+\sin^{2}\frac{j\pi}{2m}}=\frac{mQ_{2m}\left(x\right)-xP_{2m}\left(x\right)}{x\left(x^{2}+1\right)P_{2m}\left(x\right)}+\frac{1}{x^{2}+1}.$

###### Proof.

If we substitute $2x$ instead of $x$ in (4.2), we derive

$\sum_{j=0}^{2m-1}\frac{1}{x^{2}+\sin^{2}\frac{j\pi}{2m}}=\frac{2mL_{2m}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}.$

Using the well-known identity (see [22, identities (31.1) and (31.2)])

$L_{n}\left(x\right)=F_{n+1}\left(x\right)+F_{n-1}\left(x\right)$

and the defining recurrence formula (1.1) for the Fibonacci polynomials, we
deduce

$\displaystyle\sum_{j=0}^{2m-1}\frac{1}{x^{2}+\sin^{2}\frac{j\pi}{2m}}=\frac{2m\left(F_{2m+1}\left(2x\right)+F_{2m-1}\left(2x\right)\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}$
$\displaystyle=\frac{\left(2m-1\right)F_{2m+1}\left(2x\right)+\left(2m+1\right)F_{2m-1}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}+\frac{F_{2m+1}\left(2x\right)-F_{2m-1}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}$
$\displaystyle=\frac{\left(2m-1\right)F_{2m+1}\left(2x\right)+\left(2m+1\right)F_{2m-1}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}+\frac{2xF_{2m}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}$
$\displaystyle=\frac{\left(2m-1\right)F_{2m+1}\left(2x\right)+\left(2m+1\right)F_{2m-1}\left(2x\right)}{2x\left(x^{2}+1\right)F_{2m}\left(2x\right)}+\frac{1}{x^{2}+1}.$

Hence, the identity (5.1) obviously holds.

The identity (5.2) is a direct consequence of (4.2).

Finally, the identity (5.3) is derived by substituting $x$ by $2x$ in (5.2)
and using (2.21). ∎

######  Remark 5.2.

H. J. Seiffert ([28]) proposed the following formula

$\sum_{j=1}^{m-1}\frac{1}{x^{2}+\sin^{2}\frac{j\pi}{2m}}=\frac{\left(2m-1\right)F_{2m+1}\left(2x\right)+\left(2m+1\right)F_{2m-1}\left(2x\right)}{4x\left(x^{2}+1\right)F_{2m}\left(2x\right)}-\frac{1}{2x^{2}},$

which was proved by P. S. Bruckman [5] and Koshy [22, Example 40.2].
Identities (5.1)-(5.3) can be derived from this formula. Our aim was to
provide a different proof.

###  5.2. Some identities involving Fibonacci and Lucas numbers

The following two corollaries involving Fibonacci, Lucas, Pell and Pell-Lucas
numbers follow from the corresponding identities involving respective
polynomials.

######  Corollary 5.3.

The following identities hold true for any positive integer $m$

(5.4)
$\displaystyle\sum_{j=1}^{2m-1}\frac{1}{\frac{1}{4}+\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=$
$\displaystyle\frac{4}{5}\frac{\left(2m-1\right)F_{2m+1}+\left(2m+1\right)F_{2m-1}}{F_{2m}}-\frac{16}{5},$
(5.5) $\displaystyle\sum_{j=0}^{2m-1}\frac{1}{1+4\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=$ $\displaystyle\frac{2mL_{2m}}{5F_{2m}},$ (5.6)
$\displaystyle\sum_{j=0}^{2m-1}\frac{1}{1+\sin^{2}\frac{j\pi}{2m}}$
$\displaystyle=$ $\displaystyle\frac{mQ_{2m}}{P_{2m}}.$

###### Proof.

The first identity is derived by substituting $x=\frac{1}{2}$ in (5.1) and
using that $F_{n}=F_{n}\left(1\right)$. The identity (5.5) follows by
substituting $x=1$ in (5.2). Equation (5.6) is deduced from (5.3) by
substituting $x=1$, using that $P_{m}=P_{m}\left(1\right)$ and
$2Q_{m}=Q_{m}\left(1\right)$ and performing trivial calculations. ∎

The following corollary follows by combining the above results with the
identity (3.2).

######  Corollary 5.4.

The following identities hold true for any positive integer $m$:

(5.7) $2P_{2m}P_{4m}=Q_{2m}\left(Q_{4m}-1\right),$ (5.8)
$\left(2m-1\right)F_{2m+1}+\left(2m+1\right)F_{2m-1}+F_{2m}=\frac{10mF_{4m}F_{2m}}{L_{4m}-2},$
(5.9)
$\left(2m-1\right)F_{2m}F_{2m+1}+\left(2m+1\right)F_{2m-1}F_{2m}+F_{2m}^{2}=2mF_{4m}.$

###### Proof.

Setting $s=2$, $\ell=0$ and substituting $m$ by $2m$ in (3.2), and using
(2.22) we obtain that

$\sum_{j=0}^{2m-1}\frac{1}{1+\sin^{2}\frac{\pi
j}{m}}=\frac{4mU_{2m-1}\left(3\right)}{T_{2m}\left(3\right)-1}=\frac{2mP_{4m}}{Q_{4m}-1}.$

Comparing this with (5.6) yields the identity (5.7).

Setting $s=\frac{1}{2}$, $\ell=0$ and substituting $m$ by $2m$ in (3.2), we
obtain

$\frac{1}{2m}\sum_{j=0}^{2m-1}\frac{1}{\frac{1}{2}+2\sin^{2}\frac{\pi
j}{2m}}=\frac{U_{2m-1}\left(\frac{3}{2}\right)}{T_{2m}\left(\frac{3}{2}\right)-1},$

where we use that $U_{-1}\left(x\right)=0$ for all $x$. Rearranging the sum on
the left-hand side of the above display and using relations (2.3) we get

$\sum_{j=1}^{2m-1}\frac{1}{\frac{1}{4}+\sin^{2}\frac{\pi
j}{2m}}=\frac{8mF_{4m}}{L_{4m}-2}-4.$

Comparing this with (5.4) we obtain

$\frac{4}{5}\frac{\left(2m-1\right)F_{2m+1}+\left(2m+1\right)F_{2m-1}}{F_{2m}}-\frac{16}{5}=\frac{8mF_{4m}}{L_{4m}-2}-4.$

It is now straightforward to show that the last identity implies (5.8).
Substituting the well-known formula $L_{4m}=5F_{2m}^{2}+2$ (see e.g. [21,
formula 15 on p. 109]) into (5.8), we obtain (5.9). ∎

###  5.3. Some identities involving Fibonacci hyperbolic functions

In this section, by starting with (3.3), we deduce identities involving
Fibonacci and Lucas numbers and Fibonacci hyperbolic sine and cosine
functions. We prove the following proposition.

######  Proposition 5.5.

For a positive integer $m$, and positive integer $\ell<m/2,$ the following
identities hold true:

  * i)

for $m$ divisible by $4$ and $\ell$ odd

$\frac{F_{\frac{3m}{2}-3l-1}}{F_{\frac{3m}{2}}}=\frac{L_{3m-3\ell}+L_{3\ell}}{L_{3m}-2};$

  * ii)

for $m$ divisible by $4$ and $\ell$ even

$\frac{\mathrm{cFh}\left(\frac{3m-6l-2}{4}\right)}{F_{\frac{3m}{2}}}=\frac{\sqrt{5}\left(F_{3m-3\ell}+F_{3\ell}\right)}{L_{3m}-2};$

  * iii)

for $m$ even and not divisible by $4$ and $\ell$ even

$\frac{F_{\frac{3m}{2}-3l-1}}{\mathrm{sFh}\left(\frac{3m}{4}\right)}=\frac{\sqrt{5}\left(F_{3m-3\ell}+F_{3\ell}\right)}{L_{3m}-2};$

  * iv)

$\frac{\mathrm{cFh}\left(\frac{3m-6l-2}{4}\right)}{\mathrm{sFh}\left(\frac{3m}{4}\right)}=\left\\{\begin{array}[]{ll}\frac{\sqrt{5}F_{3m-3\ell}+L_{3\ell}}{\sqrt{5}F_{3m}-2},&m\text{
{not divisible by}}{\ 4},\ell\text{ odd}\\\
\frac{L_{3m-3\ell}+\sqrt{5}F_{3\ell}}{\sqrt{5}F_{3m}-2},&m\text{ odd,
}\ell\text{ even}.\end{array}\right.$

###### Proof.

We consider (3.3) when $\lambda=3\log\phi$, where $\phi$ is golden ratio.
Since $\cosh(3\log\phi)=\sqrt{5}$ and $\sinh(3\log\phi)=2$, the left-hand side
of (3.3) becomes

$\displaystyle\frac{1}{m}\sum_{j=0}^{m-1}\frac{\cos\left(2\ell\frac{j\pi}{m}\right)}{\sqrt{5}-\cos\left(2\frac{j\pi}{m}\right)}=\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\ell\frac{ij\pi}{m}}+e^{-2\ell\frac{ij\pi}{m}}}{2\left(\sqrt{5}-\cos\left(2\frac{j\pi}{m}\right)\right)}$
$\displaystyle=\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\ell\frac{ij\pi}{m}}+e^{2\ell\frac{i\left(m-j\right)\pi}{m}}}{2\left(\sqrt{5}-\cos\left(2\frac{j\pi}{m}\right)\right)}=\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\ell\frac{ij\pi}{m}}}{\sqrt{5}-\cos\left(2\frac{j\pi}{m}\right)}=\frac{1}{m}\sum_{j=1}^{m-1}\frac{e^{2\pi
i\frac{j\ell}{m}}}{\sqrt{5}-1+2\sin^{2}\frac{\pi j}{m}}.$

By applying (3.2) and (2.14) we get

(5.10) $\displaystyle\frac{1}{m}\sum_{j=1}^{m-1}\frac{e^{2\pi
i\frac{j\ell}{m}}}{\sqrt{5}-1+2\sin^{2}\frac{\pi j}{m}}$ $\displaystyle=$
$\displaystyle\frac{U_{m-\ell-1}\left(\sqrt{5}\right)+U_{\ell-1}\left(\sqrt{5}\right)}{T_{m}\left(\sqrt{5}\right)-1}$
(5.15) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\sqrt{5}\left(F_{3m-3\ell}+F_{3\ell}\right)}{2L_{3m}-4},&m,\ell\text{
even}\\\ \frac{L_{3m-3\ell}+L_{3\ell}}{2L_{3m}-4},&m\text{ even, }\ell\text{
odd}\\\ \frac{\sqrt{5}F_{3m-3\ell}+L_{3\ell}}{2\sqrt{5}F_{3m}-4},&m,\ell\text{
odd}\\\ \frac{L_{3m-3\ell}+\sqrt{5}F_{3\ell}}{2\sqrt{5}F_{3m}-4},&m\text{ odd,
}\ell\text{ even}\end{array}\right..$

The right-hand side of (3.3) for $\lambda=3\log\phi$ can be expressed by
Fibonacci hyperbolic functions (see (2.23)) as

$\frac{\cosh\left[\left(\frac{3m}{2}-3\ell\right)\log\phi\right]}{2\sinh\frac{3m\log\phi}{2}}=\frac{\mathrm{cFh}\left(\frac{3m-6\ell-2}{4}\right)}{2\
\mathrm{sFh}\left(\frac{3m}{4}\right)}.$

Using (2.24), we get following special cases

(5.16) $\frac{\mathrm{cFh}\left(\frac{3m-6\ell-2}{4}\right)}{2\
\mathrm{sFh}\left(\frac{3m}{4}\right)}=\left\\{\begin{array}[]{ll}\frac{F_{\frac{3m}{2}-3l-1}}{2F_{\frac{3m}{2}}},&m\equiv
0\pmod{4}\text{ and }\ell\text{ odd}\\\
\frac{\mathrm{cFh}\left(\frac{3m-6l-2}{4}\right)}{2F_{\frac{3m}{2}}},&m\equiv
0\pmod{4}\text{ and }\ell\text{ even}\\\
\frac{F_{\frac{3m}{2}-3l-1}}{2\mathrm{sFh}\left(\frac{3m}{4}\right)},&{m}\text{
{even, not divisible by }}{4\ }\text{{and}}{\ \ell\
}\text{{even}}\end{array}\right..$

Combining (5.10) and (5.16) we see that i)-iv) hold. ∎

######  Remark 5.6.

If we substitute $\lambda=\log\phi$ in formula (3.3) and using (2.9), we
obtain the same result as in Proposition 5.5 with $3m$ replaced by $m$ and
$3\ell$ replaced by $\ell$.

###  5.4. Identities involving Bejaia and Pisa numbers

In this section, we derive some further identities involving Chebyshev
polynomials, Fibonacci and Lucas numbers and the so-called Bejaia and Pisa
numbers which are generalizations of the bisected Fibonacci and Lucas numbers
respectively. For a positive integer $N$, Bejaia and Pisa numbers
$\mathcal{B}_{\ell}(N)$ and $\mathcal{P}_{\ell}(N)$, respectively were
introduced by Chair [10, relations (17) and (18)] and defined as:

$\displaystyle\mathcal{B}_{\ell}(N)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N\left(N-4\right)}}\left(\left(\frac{N-2+\sqrt{N\left(N-4\right)}}{2}\right)^{\ell}-\left(\frac{N-2-\sqrt{N\left(N-4\right)}}{2}\right)^{\ell}\right),$
$\displaystyle\mathcal{P}_{\ell}(N)$ $\displaystyle=$
$\displaystyle\left(\frac{N-2+\sqrt{N\left(N-4\right)}}{2}\right)^{\ell}+\left(\frac{N-2-\sqrt{N\left(N-4\right)}}{2}\right)^{\ell}.$

Those numbers are related to a trigonometric sum similar to (4.4) and occur in
the (exact) formula for the two-point resistance of the complete graph minus
$N$ edges of the opposite vertices, for $N$ odd. Namely, Chair in [10,
relation (25)] derived the following formula, valid for $\ell<\frac{N+1}{2}$:

(5.17)
$\frac{4}{N}\sum\limits_{n=0}^{\frac{N-1}{2}}\frac{\sin^{2}\left(\frac{2n\ell\pi}{N}\right)}{N-4\sin^{2}\left(\frac{n\pi}{N}\right)}=\frac{\mathcal{B}_{2\ell}(N)}{2}-\frac{\sqrt{N\left(N-4\right)}}{2}\mathcal{B}_{\ell}^{2}(N)D_{N},$

where

$D_{N}=\left(1-\left(\frac{N-2-\sqrt{N\left(N-4\right)}}{2}\right)^{N}\right)\left(1+\left(\frac{N-2-\sqrt{N\left(N-4\right)}}{2}\right)^{N}\right)^{-1}.$

As above, it is possible to express the trigonometric sum on the left-hand
side of (5.17) in terms of Chebyshev polynomials and deduce the following
corollary.

######  Corollary 5.7.

For positive integers $N$ and $\ell<\frac{N+1}{2}$ the following identity
holds true

(5.18)
$\frac{4}{N}\sum\limits_{n=0}^{\frac{N-1}{2}}\frac{\sin^{2}\left(\frac{2n\ell\pi}{N}\right)}{N-4\sin^{2}\left(\frac{n\pi}{N}\right)}=\\\
=\frac{U_{N-2\ell-1}\left(-\frac{N}{2}+1\right)+U_{2\ell-1}\left(-\frac{N}{2}+1\right)-U_{N-1}\left(-\frac{N}{2}+1\right)}{2T_{N}\left(-\frac{N}{2}+1\right)-2}.$

###### Proof.

The proof is analogous to the proof of equation (4.5), with the difference
that in this case, we take $s=-N/2$ in (3.2). We omit the details. ∎

By specifying $N$ in (5.18), and combining it with (5.17), one can produce
many interesting identities. In the following examples we present two
possibilities.

######  Example 5.8.

Let us take $N=9$ and $\ell\in\\{0,1,2,3,4\\}$ in (5.18). Then, (5.18) and
(5.17) for $N=9$ yield the identity

$\frac{\mathcal{B}_{2\ell}(9)}{2}-\frac{3\sqrt{5}}{2}\mathcal{B}_{\ell}^{2}(9)D_{9}=-\frac{1}{2}\left(\frac{U_{8}\left(-\frac{7}{2}\right)-U_{8-2\ell}\left(-\frac{7}{2}\right)-U_{2\ell-1}\left(-\frac{7}{2}\right)}{T_{9}\left(-\frac{7}{2}\right)-1}\right).$

Using symmetry properties (2.1) and (2.2), together with relations (2.4) we
get

$\frac{\mathcal{B}_{2\ell}(9)}{2}-\frac{3\sqrt{5}}{2}\mathcal{B}_{\ell}^{2}(9)D_{9}=\frac{1}{3}\frac{-F_{36-8\ell}+F_{8\ell}+F_{36}}{L_{36}+2}.$

Therefore, we obtain the following identity

$2\left(F_{8\ell}+F_{36}-F_{36-8\ell}\right)=\left(3\mathcal{B}_{2\ell}(9)-9\sqrt{5}\mathcal{B}_{\ell}^{2}(9)D_{9}\right)\left(L_{36}+2\right).$

######  Example 5.9.

In this example we take $N=L_{2k}+2$, where $k$ is such that $L_{2k}$ is odd
and take $\ell<\frac{N+1}{2}$. By reasoning analogously as in the previous
example, using special values (2.15) and (2.16) of Chebyshev polynomials, we
derive the following identity

$\frac{F_{2Nk}+F_{2\ell
k}-F_{2\left(N-2\ell\right)k}}{\left(L_{2Nk}+2\right)F_{2k}}=\frac{1}{2}\left(\mathcal{B}_{2\ell}(N)-\sqrt{N\left(N-4\right)}\mathcal{B}_{\ell}^{2}(N)D_{N}\right).$

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