CLOSED EXPRESSIONS OF THE FIBONACCI POLYNOMIALS IN TERMS OF TRIDIAGONAL DETERMINANTS

In the paper, the authors find a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

In mathematics, a closed form is a mathematical expression that can be evaluated in a finite number of operations.It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit.
It is general knowledge 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 [26,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 [35], the Euler numbers E 2k were represented by lower Hessenberg determinants In [34,Theorem 1.1], the Euler numbers E 2k for k ∈ N were represented by lower Hessenberg and sparse 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 [2,3,4,6,7,8,9,10,11,12,13,18,19,20,21,23,25,26,27,28,29,31,34] 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 with The Fibonacci numbers F n can be viewed as a particular case F n (1) of the Fibonacci polynomials which can be generated by See the monograph [5], the websites [32,33], and related references therein.In [17, p. 215, Example 1], it was deduced that , n ∈ N.
In [15,16], among other things, it was listed that for r ∈ N, where In this paper, motivated by the above closed 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 new closed 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.For n ∈ N, the Fibonacci polynomials F n (s) can be expressed as

Proof of Theorem 1
Now we start out to concisely prove Theorem 1 by virtue of the formula (5).Applying u(t) = t and v(t) = 1 − ts − t 2 to the formula (5) yields as t → 0 for n ∈ N. By the generating function in (2), we obtain that which can be rewritten as the expression (3).The expression (4) follows from taking the limit s → 1 on both sides of the expression (3).The proof of Theorem 1 is complete.

Remarks
Finally, we list several remarks on Lemma 1, Theorem 1, and others as follows.
Remark 1.As showed, for example, in recently published papers [24,26,30,34] and in this paper, the formula (5) in Lemma 1 is an effectual tool to express some quantities in mathematics as lower Hessenberg determinants.
Remark 2. It is easy to see that, from the expressions (3) and ( 4), we can recover the recurrence relation F n (s) = sF n−1 (s) + F n−2 (s) in [33] and the recurrence relation (1) for n ≥ 3.
Remark 3. The expressions ( 3) and ( 4) can be rearranged as Remark 4. The Bernoulli numbers B n , the Bernoulli polynomials B n (u), the Euler numbers E n , and the Euler polynomials E n (x) can be expressed as different closed forms.Let S(n, m) stand for the Stirling numbers of the second kind which can be computed by In [26, Theorem 1.1], the Bernoulli polynomials B n (u) were expressed as In mathematics, different forms have different implications.In other words, from different forms of mathematical quantities, one can read out different meanings and significance.Therefore, expressing the Fibonacci numbers F n and the Fibonacci polynomials F n (s) in terms of tridiagonal determinants is meaningful and significant.