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Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants

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Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 CLOSED EXPRESSIONS OF THE FIBONACCI POLYNOMIALS IN TERMS OF TRIDIAGONAL DETERMINANTS FENG QI Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China JING-LIN WANG Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China BAI-NI GUO School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Abstract. 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. 1. Main results A tridiagonal matrix (determinant) is a square matrix (determinant) with nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal. See the paper [14] and closely-related references therein. A square matrix H = (hij)n×n is called a tridiagonal matrix if hij = 0 for all pairs (i; j) such that ji − jj > 1. On the other hand, a matrix H = (hij)n×n is called a lower (or an upper, respectively) Hessenberg matrix if hij = 0 for all pairs (i; j) such that i + 1 < j (or j + 1 < i, respectively). See the paper [22] and closely-related references therein. E-mail addresses: [email protected], [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. 11B39, 11B83, 11C20, 11Y55, 15A15, 65F40. Key words and phrases. closed expression; Fibonacci number; Fibonacci polynomial; tridiagonal determinant; Hessenberg determinant. This paper was typeset using AMS-LATEX. 1 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 2 F. QI, J.-L. WANG, AND B.-N. GUO In mathematics, a closed form is a mathematical expression that can be evaluated in a finite num- ber 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 Bk and Bk(u) for k ≥ 0 satisfy Bk(0) = Bk and can be generated respectively by 1 1 1 z X zk z X z2k zeuz X zk = B = 1 − + B and = B (u) ez − 1 k k! 2 2k (2k)! ez − 1 k k! k=0 k=1 k=0 x x for jzj < 2π. Because the function ex−1 − 1 + 2 is odd in x 2 R, all of the Bernoulli numbers B2k+1 for k 2 N equal 0. In [26, Theorem 1.2], the Bernoulli polynomials Bk(u) for k 2 N were expressed by lower Hessenberg determinants k 1 ` + 1 `−m+1 `−m+1 Bk(u) = (−1) (1 − u) − (−u) : ` + 1 m 1≤`≤k;0≤m≤k−1 Consequently, the Bernoulli numbers Bk for k 2 N can be represented by lower Hessenberg deter- minants k 1 ` + 1 Bk = (−1) : ` + 1 m 1≤`≤k;0≤m≤k−1 For k 2 f0g [ N and x 2 R, the Euler numbers Ek and the Euler polynomials Ek(x) can be generated respectively by t=2 1 k xt 1 2e X Ek t 2e X Ek(x) = and = tk et + 1 k! 2 et + 1 k! k=0 k=0 2et=2 for t 2 (−π; π). Since the generating function et+1 of the Euler numbers Ek is even on (−π; π), then E2k−1 = 0 for all k 2 N. At the website [35], the Euler numbers E2k were represented by lower Hessenberg determinants 1 2! 1 0 ··· 0 0 1 1 4! 2! 1 ··· 0 0 k . .. E2k = (−1) (2k)! . ; k 2 N: 1 1 1 1 (2k−2)! (2k−4)! (2k−6)! ··· 2! 1 1 1 1 1 1 (2k)! (2k−2)! (2k−4)! ··· 4! 2! In [34, Theorem 1.1], the Euler numbers E2k for k 2 N were represented by lower Hessenberg and sparse determinants k i π E2k = (−1) cos (i − j + 1) : j − 1 2 (2k)×(2k) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 CLOSED EXPRESSIONS OF FIBONACCI POLYNOMIALS 3 Recently, the first author represented the Euler polynomials Ek(x) for k ≥ 0 by lower Hessenberg determinants 1 2 0 0 ··· 0 0 0 1 x 0 2 0 ··· 0 0 0 2 2 2 x 0 1 2 ··· 0 0 0 3 3 3 3 k x ··· 0 0 0 (−1) 0 1 2 Ek(x) = . : 2k . .. xk−2 k−2 k−2 k−2 ··· k−2 2 0 0 1 2 k−3 k−1 k−1 k−1 k−1 k−1 k−1 x 0 1 2 ··· k−3 k−2 2 k k k k k k k x 0 1 2 ··· k−3 k−2 k−1 Consequently, the Euler numbers Ek for k ≥ 0 can be expressed by lower Hessenberg determinants 1 2 0 0 ··· 0 0 0 1 1 2 0 2 0 ··· 0 0 0 1 2 2 22 0 1 2 ··· 0 0 0 1 3 3 3 3 ··· 0 0 0 k 2 0 1 2 Ek = (−1) . : . .. 1 k−2 k−2 k−2 ··· k−2 2 0 2k−2 0 1 2 k−3 1 k−1 k−1 k−1 k−1 k−1 2k−1 0 1 2 ··· k−3 k−2 2 1 k k k k k k 2k 0 1 2 ··· k−3 k−2 k−1 For more and detailed information on the Bernoulli numbers Bk, the Bernoulli polynomials Bk(u), the Euler numbers Ek, and the Euler polynomials Ek(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 p n p n 1 + 5 − 1 − 5 Fn = p 2n 5 for n 2 N form a sequence of integers and satisfy the linear recurrence relation Fn = Fn−1 + Fn−2 (1) with F1 = F2 = 1. The first fourteen Fibonacci numbers Fn for 1 ≤ n ≤ 14 are 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377: The Fibonacci numbers Fn can be viewed as a particular case Fn(1) of the Fibonacci polynomials p n p n 1 s + 4 + s2 − s − 4 + s2 Fn(s) = p 2n 4 + s2 which can be generated by 1 t X = F (s)tn = t + st2 + s2 + 1t3 + s3 + 2st4 + ··· : (2) 1 − ts − t2 n n=1 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 4 F. QI, J.-L. WANG, AND B.-N. GUO See the monograph [5], the websites [32, 33], and related references therein. In [17, p. 215, Exam- ple 1], it was deduced that 1 −1 0 0 ··· 0 0 1 1 −1 0 ··· 0 0 0 1 1 −1 ··· 0 0 F = ; n 2 : n . .. N . 0 0 0 0 ··· 1 −1 0 0 0 0 ··· 1 1 (n−1)×(n−1) In [15, 16], among other things, it was listed that x 1 0 ··· 0 0 1 1 1 ··· 1 1 −1 x 1 ··· 0 0 −1 0 1 ··· 1 1 0 −1 x ··· 0 0 0 −1 0 ··· 1 1 F (x) = ;F = ; r+1 . .. r−1 . .. 0 0 0 ··· x 1 0 0 0 ··· 0 1 0 0 0 · · · −1 x 0 0 0 · · · −1 0 r×r r×r 1 2 3 ··· r − 1 r −1 1 2 ··· r − 2 r − 1 0 −1 1 ··· r − 3 r − 2 F = jq j ;F = ;F = jQ j r−1 ij r×r 2r . .. 2r+1 ij r×r . 0 0 0 ··· 1 2 0 0 0 · · · −1 1 r×r for r 2 N, where 8 8 −1; i = j + 1; −1; i − j + 1; > <> <>2; i = j; qij = i + j + 1 mod 2; i ≤ j; and Qij = > >1; i < j; :0; otherwise > :0; otherwise: In this paper, motivated by the above closed expressions for the Bernoulli numbers Bk, the Bernoulli polynomials Bk(u), the Euler numbers Ek, the Euler polynomials Ek(x), the Fibonacci numbers Fn, and the Fibonacci polynomials Fn(s), we will find a new closed expression for the Fibonacci polynomials Fn(s) and, consequently, for the Fibonacci numbers Fn, in terms of a tridi- agonal determinant. The main results of this paper can be stated as the following theorem. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 CLOSED EXPRESSIONS OF FIBONACCI POLYNOMIALS 5 Theorem 1. For n 2 N, the Fibonacci polynomials Fn(s) can be expressed as 0 −1 0 0 ··· 0 0 0 1 s −1 0 ··· 0 0 0 0 2 · 1 2s −1 ··· 0 0 0 1 0 0 3 · 2 3s ··· 0 0 0 F (s) = (3) n .
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  • 2-Fibonacci Polynomials in the Family of Fibonacci Numbers
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