The graded generalized Fibonacci sequence and Binet formula

Transcription

1 The graded generaized Fibonacci sequence and Binet formua Won Sang Chung,, Minji Han and Jae Yoon Kim Department of Physics and Research Institute of Natura Science, Coege of Natura Science, Gyeongsang Nationa University, Jinju , Korea E-mais: Key Word: MSC number: Abstract In this paper the graded generaized Fibonacci numbers are introduced as a kind of generaization of the Fibonacci numbers. Some properties for the graded generaized Fibonacci numbers are investigated. Key Word: graded generaized Fibonacci number, Binet formua. MSC number: 11B39. 1

2 1 Introduction The Fibonacci sequence is a series of numbers, starting with the two succeeding integers 0 and 1, the vaue of each eement is given by the sum of the two preceding it. It appears in various appication such as Pasca s triange [1], computer agorithms [, 3] and graph theory [4, 5]. The ordinary Fibonacci sequence is given by a n = a n 1 + a n, n, with a 0 = 0, a 1 = 1 (1) There has been severa types of generaization of Fibonacci sequence [6]. The Lucas sequence is defined by the same recurrence reation with a 0 =, a 1. The Jacobstha sequence [7] is defined by the recurrence reation a n = a n 1 + a n, n with a 0 = 0, a 1 = 1 and the Jacobstha- Lucas sequence [6] is defined by the same recurrence reation with a 0 =, a 1. B. Singh, O. Sikhwa and S. Bhatnagar [8] defined Fibonacci-ike sequence a n = a n 1 + a n, n with a 0 =, a 1 =. In genera, Horadam [9] defined the generaized Fibonacci sequence a n = a n 1 + a n, n 3 with a 1 = p, a = p + q. Kaman and Mena [10] generaized the Fibonacci sequence by a n = aa n 1 + ba n, n 3 with a 0 = 0, a 1 = 1 p and q are arbitrary integers. In this paper we discuss a kind of generaization of the Fibonacci number which we ca the graded generaized Fibonacci number F n. Here, grading impies the insertion of the factor ( 1) n in the origina recurrence reation. This factor ( 1) n is interpreted as the parity operation in the Fock space corresponding to the quantum mechanics. For a ordinary boson agebra, the parity was first introduced by Wigner [11]. The ordinary boson agebra is reated to the ordinary Hermite sequence, but Wigner s agebra is reated to the graded Hermite sequence whose recurrence reation contains the factor ( 1) n [11-14]. For the graded generaized Fibonacci number, we construct the even generating function and odd generating function for the graded generaized Fibonacci numbers and find some properties of the graded generaized Fibonacci number. Graded generaized Fibonacci number We now introduce a further generaization of the Fibonacci numbers; we sha ca it the graded generaized Fibonacci sequence. Definition.1 The graded generaized Fibonacci sequence is defined through the foowing recurrence reation: F n = (µ + µ ( 1) n )F n 1 + (ν + ν ( 1) n )F n (n ) () with F 0 = 0, F 1 = 1, µ, ν, µ, ν N.

3 The first few graded Fibonacci numbers are: F 0 = 0 F 1 = 1 F = µ + F 3 = µ + µ + ν F 4 = µ + (µ + µ + ν + + ν ) F 5 = (µ + µ ) + (ν + + ν )µ + µ + ν F 6 = µ + [(µ + µ ) + (ν + + ν )µ + µ + ν (ν + + ν )], (3) µ ± = µ ± µ, ν ± = ν ± ν (4) When µ = ν = 0, we have an ordinary generaized Fibonacci sequence. From the recurrence reation (), we have the foowing couped recurrence reations: F m = µ + F m 1 + ν + F m (m 1), F m+1 = µ F m + ν F m 1 (m 1), (5) For these two recurrence reations, et us define the even generating function g e and odd generating function g o as foows: g e (t) = F m t m (6) g o (t) = m=0 F m+1 t m+1 (7) n=0 Then, we have the foowing couped equations: Soving this system, we have g o = g e = g e = µ + tg o + ν + t g e g o = µ tg e + ν t g o + t (8) t(1 ν + t ) (1 ν + t )(1 ν t ) µ + µ t (9) µ + t (1 ν + t )(1 ν t ) µ + µ t (10) Thus, the generating function for the graded generaized Fibonacci poynomia is given by g(t) = g e + g o = t(1 + µ + t ν + t ) (1 ν + t )(1 ν t ) µ + µ t = F n t n (11) n=0 3

4 Proposition.1 The graded generaized Fibonacci poynomia can be expressed in terms of series: F n+1 = n 1 n 1 F n = µ + ( ν + ν ) (µ + µ + ν + + ν ) n 1, (1) n n n 1 ( ν + ν ) (µ + µ +ν + +ν ) n n 1 ( ν + ν ) (µ + µ +ν + +ν ) n 1, (13) n 1. Proof. From the eq.(10), we have g e = µ + t 1 t (µ + µ + ν + + ν ν + ν t ) = µ + t t m (µ + µ + ν + + ν ν + ν t ) m m 0 n 1 n 1 = µ + (µ + µ + ν + + ν ) n 1 ( ν + ν ) t n n 0 (14) which competes the proof of the eq.(1). The eq.(13) is simiary obtained from the eq.(9). 3 Binet Formua From the recurrence reation (5), we have F n+ = (µ + µ + ν + + ν )F n ν + ν F n (15) Soving the above recurrence reation, we have F n = µ + [n] φ,ψ, (16) and [n] φ,ψ = φn ψ n φ ψ φ = µ +µ + ν + + ν + (µ + µ + ν + + ν ) 4ν + ν (17) (18) ψ = µ +µ + ν + + ν (µ + µ + ν + + ν ) 4ν + ν Inserting the eq.(16) into the eq.(5), we have (19) F n+1 = [n + 1] φ,ψ ν + [n] φ,ψ (0) 4

5 Here, we demand that (µ + µ + ν + + ν ) 4ν + ν. From the eq.(16) and the eq.(0), we know F n+1 im = φ ν + = µ +µ ν + (µ+ µ + ν + + ν ) 4ν + ν (1) n F n µ + µ + F n+ im = µ +φ = µ +µ + ν + (µ+ µ + ν + + ν ) 4ν + ν () n F n+1 φ ν + µ In genera, the above two vaues are different so the graded generaized sequence osciates, which impies that this sequence does not converge. But, if four parameters µ, ν, µ, ν obey the reation (ν µ )µ (µ µ ) = ν (µ 1), (3) the graded generaized sequence converges to φ ν + µ +, which we can ca a graded generaized goden ratio. Now et us consider the foowing imit: F n+ F n+3 im = φ, im = φ (4) n F n n F n+1 Thus, two subsequence {F 0, F, F 4, } and {F 1, F 3, F 5, } aways converge to φ. From the eq.(18) and the eq.(19), we can easiy find q = φ = 1 ( µ + ν ν + ν µ + µ + ν + ) ν + ν µ (5) p = ψ = 1 ( µ + ν ν + ν µ µ + ν + ) ν + ν µ, (6) we considered the case that ν > ν. From the above reations, we have From the eq.(16) and the eq.(0), we obtain [ µ n ] + (n even) φ,ψ F n = [ n+1 ] [ n 1 ] ν+ or F n = 1 {} q,p The eq.(9) can be written as F n = qp = ν + ν (7) φ,ψ 1 + ( 1) n [µ + {n} q,p + 1 ( 1)n {n} q,p = qn ( p) n q + p (n odd) ] ({n + 1} q,p ν + {n 1} q,p ) 1 (q p ) [(1 + µ + α)q n + (1 + µ α)( q) n (1 + µ + β)p n (1 + µ β)( p) n ] (31) α = q + (8) (9) (30) ν+ ν+ p, β = p + q (3) ν ν 5

6 4 Concusion In this paper we discussed a kind of generaization of the Fibonacci number which we ca the graded generaized Fibonacci number F n. The recurrence reation for the graded generaized Fibonacci numbers takes the different forms for even n and odd n. Thus, we constructed the even generating function and odd generating function for the graded generaized Fibonacci numbers. We found some properties reated to the graded generaized Fibonacci numbers. We aso found that two subsequence {F 0, F, F 4, } and {F 1, F 3, F 5, } converge to a number φ whie the graded generaized Fibonacci sequence osciates. Acknowedgement This work was supported by the Nationa Research Foundation of Korea Grant funded by the Korean Government (NRF-015R1D1A1A ) and by the Gyeongsang Nationa University Fund for Professors on Sabbatica Leave, 016. References [1] T. Koshy, Fibonacci and Lucas Numbers with Appications, Wiey, New York, 001. [] J. Atkins, R. Geist, Fibonacci numbers and computer agorithms, Coege Math. J. 18 (1987), [3] M. L. Fredman, R. E. Tarjan, Fibonacci heaps and their uses in improved network optimization agorithms, J. ACM 34, (1987), [4] Z. R. Bogdonowicz, Formuas for the number of spanning trees in a fan, App. Math. Sci., (008), [5] P. Chebotarev, Spanning forests and the goden ratio, Discrete App. Math. 156, (008), [6] T. Koshy, Fibonacci and Lucas Numbers with Appications, A Wiey-Interscience Pubication, New York, 001. [7] A.F. Horadam, Jacobstha Representation Numbers, The Fib. Quart, 34, (1996), [8] B. Singh, O. Sikhwa and S. Bhatnagar, Fibonacci-Like Sequence and its, Properties, Int. J. Contemp. Math. Sciences, 5(18), (010), [9] A.F. Horadam, The Generaized Fibonacci Sequences, The American Math. Monthy, 68(5), (1961), [10] D. Kaman and R. Mena, The Fibonacci Numbers.Exposed, The Mathematica Magazine,, (00). [11] E.Wigner, Agebraic generaization of quantum mechanics, Phys.Rev.77, (1950). [1] C. D. Batista and G. Ortiz, Agebraic approach to interacting quantum systems, Advances in Physics, Vo. 53, No. 1, 1-8, (004). 6

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