A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant

In the paper, after concisely reviewing and surveying some known results, the authors ﬁnd a determinantal expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

is called a tridiagonal matrix if h i, j = 0 for all pairs (i, j) such that |i − j| > 1. On the other hand, a matrix H = (h i, j ) n×n is called a lower (or an upper, respectively) Hessenberg matrix if h i, j = 0 for all pairs (i, j) such that i + 1 < j (or j + 1 < i, respectively). See the paper [10] and closely related references therein. It is general knowledge [11,14] that the Bernoulli numbers and polynomials B k and B k (u) for k ≥ 0 satisfy B k (0) = B k and can be generated, respectively, by for |z| < 2π . Because the function x e x −1 − 1 + x 2 is odd in x ∈ R, all of the Bernoulli numbers B 2k+1 for k ∈ N equal 0. In [15,Theorem 1.2], the Bernoulli polynomials B k (u) for k ∈ N were expressed by lower Hessenberg determinants Consequently, the Bernoulli numbers B k for k ∈ N can be represented by lower Hessenberg determinants For k ∈ {0} ∪ N and x ∈ R, the Euler numbers E k and the Euler polynomials E k (x) can be generated, respectively, by for t ∈ (−π, π). Since the generating function 2e t/2 e t +1 of the Euler numbers E k is even on (−π, π), then E 2k−1 = 0 for all k ∈ N. At the website https://en.wikipedia.org/ wiki/Euler_number, the Euler numbers E 2k were represented by lower Hessenberg determinants In [18, Theorem 1.1], the Euler numbers E 2k for k ∈ N were represented by lower Hessenberg determinants Recently, the first author represented the Euler polynomials E k (x) for k ≥ 0 by lower Hessenberg determinants . .
Consequently, the Euler numbers E k for k ≥ 0 can be expressed by lower Hessenberg determinants For more and detailed information on the Bernoulli numbers B k , the Bernoulli polynomials B k (u), the Euler numbers E k , and the Euler polynomials E k (x), please refer to recently published papers such as [3][4][5][6]13,[15][16][17][18] and plenty of closely related references therein.
It is well known that the Fibonacci numbers for n ∈ N form a sequence of integers and satisfy the linear recurrence relation The Fibonacci numbers F n can be viewed as a particular case F n (1) of the Fibonacci polynomials which can be generated by For details on the Fibonacci numbers F n and properties, see the monograph [2] and the related references therein. In [9, p. 215, Example 1], it was deduced that for n ∈ N. In [8], among other things, it was listed that otherwise.
In this paper, motivated by the above determinantal expressions for the Bernoulli numbers B k , the Bernoulli polynomials B k (u), the Euler numbers E k , the Euler polynomials E k (x), the Fibonacci numbers F n , and the Fibonacci polynomials F n (s), we will find a determinantal expression for the Fibonacci polynomials F n (s) and, consequently, for the Fibonacci numbers F n , in terms of a tridiagonal determinant.
The main results of this paper can be stated as the following theorem.

Theorem 1.1 For n ∈ N, the Fibonacci polynomials F n (s) can be expressed as
and, consequently, the Fibonacci numbers F n can be expressed as (1.4)

A Lemma
To supply a concise proof of Theorem

Proof of Theorem 1.1
Now, we start out to concisely prove Theorem 1.1 by virtue of the formula (2.1). Applying for n ∈ N. By the generating function in (1.2) and the above identity, we obtain that

Remarks
Now, we list several remarks on Lemma 2.1, Theorem 1.1, and others as follows. Remark 4.1 As shown, for example, in recently published papers [12,15,18] and in this paper, the formula (2.1) in Lemma 2.1 is an effectual tool to express some quantities in mathematics as lower Hessenberg determinants.