#A48 INTEGERS 9 (2009), A NEW GENERALIZATION OF FIBONACCI SEQUENCE AND EXTENDED BINET S FORMULA

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1 #A48 INTEGERS 9 009), A NEW GENERALIZATION OF FIBONACCI SEQUENCE AND EXTENDED BINET S FORMULA Marcia Edson Department of Mathematics & Statistics, Murray State University, Murray, KY marcia.edson@murraystate.edu Omer Yayenie 1 Department of Mathematics & Statistics, Murray State University, Murray, KY omer.yayenie@murraystate.edu Received: 7/1/08, Revised: 5/8/09, Accepted: 8/8/09, Published: 1/18/09 Abstract Consider the Fibonacci sequence {F n} n0 with initial conditions F0 0, F1 1 and recurrence relation F n F n 1 + F n n ). The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this article, we study a new generalization {q n}, with initial conditions q 0 0 and q 1 1, which is generated by the recurrence relation q n aq n 1 + q n when n is even) or q n bq n 1 + q n when n is odd), where a and b are nonzero real numbers. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {q n} with a b 1. Pell s sequence is {q n} with a b while k-fibonacci sequence has a b k. We produce an extended Binet s formula for {q n} and, thereby, identities such as Cassini s, Catalan s, d Ocagne s, etc. 1. Introduction The Fibonacci sequence, {F n } n0, is a series of numbers, starting with the integer pair 0 and 1, where the value of each element is calculated as the sum of the two preceding it. That is, F n F n 1 +F n for all n. The first few terms of the Fibonacci sequence are: 0, 1, 1,, 3, 5, 8, 13, 1, 34, 55, 89, 144, 33, 377, 610, 987, 1597, 584,.... The Fibonacci numbers are perhaps most famous for appearing in the rabbit-breeding problem, introduced by Leonardo de Pisa in 10 in his book called Liber Abaci. However, they also occur in Pascal s triangle [18], in Pythagorean triples [18], computer algorithms [1, 9, 33], some areas of algebra [5, 8, 31], graph theory [, 3], quasicrystals [34, 41], and many other areas of mathematics. They occur in a variety of other fields such as finance, art, architecture, music, etc. See [10] for extensive resources on Fibonacci numbers.) However, in this paper, we are most interested in the generalizations of the Fibonacci sequence. Some authors [13, 15, 17, 7, 37] have generalized the Fibonacci sequence by preserving the recurrence relation and altering the first two terms of the sequence, while others [7, 0, 1,, 6, 30, 40] have generalized the Fibonacci sequence by preserving the first two terms of the sequence but altering the recurrence 1 Corresponding Author.

2 INTEGERS: 9 009) 640 relation slightly. One example of this latter generalization, called the k-fibonacci sequence, {F k,n } n0, is defined using a linear recurrence relation depending on one real parameter k) given by F k,n kf k,n 1 + F k,n n ) where F k,0 0 and F k,1 1. When k 1, the classical Fibonacci sequence is obtained. These generalizations satisfy identities that are analogous to the identities satisfied by the classical Fibonacci sequence [18]. We now introduce a further generalization of the Fibonacci sequence; we shall call it the generalized Fibonacci sequence. Unlike other variations, this new generalization depends on two real parameters used in a non-linear recurrence relation. Definition 1. { For any } two nonzero real numbers a and b, the generalized Fibonacci sequence, say, is defined recursively by F a,b) n n0 F a,b) 0 0, F a,b) 1 1, F a,b) n { af a,b) n 1 bf a,b) n 1 + F a,b) n, if n is even + F a,b) n, if n is odd n ). To avoid cumbersome notation, let us denote F n a,b) {q n } satisfies by q n. Thus, the sequence { aqn 1 + q q 0 0, q 1 1, q n n, if n is even bq n 1 + q n, if n is odd n ). We now note that this new generalization is in fact a family of sequences where each new choice of a and b produces a distinct sequence. When a b 1, we have the classical Fibonacci sequence and when a b, we get the Pell numbers. Even further, if we set a b k, for some positive integer k, we get the k-fibonacci numbers, the generalization of the Fibonacci numbers mentioned above. We will describe the terms of the sequence {q n } explicitly by using a generalization of Binet s formula. Therefore, we will start the main content of the paper by deriving a generalization of Binet s formula via generating functions) and then will present extensions of well-known Fibonacci identities such as Catalan s, Cassini s, and d Ocagne s. Later, we alter {q n } by allowing arbitrary initial conditions and also consider the convergence of the ratios of successive terms of the sequence. It is well-known that the ratios of successive Fibonacci numbers approach the golden mean, Φ, so it is natural to ask if analogous results exist for the variations and extensions of the Fibonacci sequence. Even for random Fibonacci sequences, there are results related to growth and decay rates [6, 16, 9, 36, 39]. We now give a brief word-combinatorial interpretation of the generalized Fibonacci sequence as this is the context in which we first studied this family of sequences.

3 INTEGERS: 9 009) 641 Let 0 < α < 1 be an irrational number. Associate with α a sequence, called the characteristic sequence of α see [3]), which is denoted by ω ωα), and given by ω ω 1 ω ω 3 ω n, where ω n n + 1)α nα n 1). Note that ω n {0, 1}, so that ωα) is an infinite word consisting of 0 s and 1 s. We now outline the relation between the characteristic sequence of α and the continued fraction expansion of α. This connection leads us, via word combinatorics, to the definition of the generalized Fibonacci sequences. Suppose that the continued fraction expansion of α [0; 1 + d 1, d, d 3,...], and define a sequence {s n } n 0 of words by s 0 1, s 1 0, and s n s dn 1 n 1 s n, n ). Then for n 1, each s n is a prefix of ωα) and ωα) lim n s n see [3]). Example. The infinite Fibonacci word) Let α [0;, 1, 1, 1,...] 1 φ, where φ is the Golden Mean. Then the {s n } are s 0 1, s 1 0, s 01, s 3 010, s , s ,.... The limit of this sequence is the infinite word, ω Since the lengths of s 0 and s 1 are both 1 and s n is obtained by concatenating s n 1 and s n, the length of the word s n, denoted by s n, is F n+1, the n + 1 st Fibonacci number. Since the lengths of these subwords are Fibonacci numbers, the infinite word ω is called the Fibonacci word. { Now we associate with the generalized Fibonacci sequence, F a,b) n } {q n }, a unique quadratic irrational number α in the interval 0, 1), whose continued fraction expansion has the form α [0; a, b, a, b,...] [0; a, b, a, b]. Then ωα) lim n s n where and s 0 1, s 1 0, s 0 a , { s a s n n 1 s n, if n is even s b n 1s n, if n is odd n 3). We next define a number sequence {r n } as follows. Let r 0 0, r n s n n 1). Since {r n } and {q n } satisfy the same initial conditions and have the same recursive definitions, clearly {r n } {q n }.

4 INTEGERS: 9 009) 64 In conclusion, every generalized Fibonacci sequence with a and b nonnegative integers has a one-to-one correspondence with a quadratic irrational α in the interval 0, 1) having the form α [0; a, b, a, b,...]. Moreover, every generalized Fibonacci sequence is intimately connected to an infinite word called the characteristic sequence of α. Example 3. Let α [0; 1, 1, 1, 1,...] 1 φ, where φ is the Golden Mean. Then the terms of the sequence {s n } are s 0 1, s 1 0, s 1, s 3 10, s 4 101,.... Observe that {r n } {F n }, the Fibonacci sequence and that ωα) can be obtained from the infinite Fibonacci word by exchanging 0 s and 1 s.. Generating Function for the Generalized Fibonacci Sequence Generating functions provide a powerful technique for solving linear homogeneous recurrence relations. Even though generating functions are typically used in conjunction with linear recurrence relations with constant coefficients, we will systematically make use of them for linear recurrence relations with nonconstant coefficients. In this section, we consider the generating functions for the generalized Fibonacci sequences and derive some of the most fascinating identities satisfied by these sequences. As Wilf indicated in [38], a generating function is a clothesline on which we hang up a sequence of numbers for display. Theorem 4. The generating function for the generalized Fibonacci sequence {q n } is F x) x 1 + ax x ) 1 ab + )x + x 4. Proof. We begin with the formal power series representation of the generating function for {q n }, F x) q 0 + q 1 x + q x + + q k x k + q m x m. Note that, bxf x) bq 0 x+bq 1 x +bq x 3 + +bq k x k+1 + bq m x m+1 bq m 1 x m m0 m0 m1

5 INTEGERS: 9 009) 643 and x F x) q 0 x + q 1 x 3 + q x q k x k+ + q m x m+ q m x m. m0 m Since q k+1 bq k + q k 1 and q 0 0, q 1 1, we get 1 bx x ) F x) x + q m bq m 1 q m ) x m. m1 Since q k aq k 1 + q k, we get 1 bx x ) F x) x + a b) q m 1 x m m1 1 bx x ) F x) x + a b) x q m 1 x m 1. m1 Now let Since fx) q m 1 x m 1. m1 q k+1 bq k + q k 1 b aq k 1 + q k ) + q k 1 ab + 1) q k 1 + bq k ab + 1) q k 1 + q k 1 q k 3 ab + ) q k 1 q k 3, we have 1 ab + )x + x 4) fx) x x 3 + q m 1 ab + )q m 3 + q m 5 ) x m 1 x x 3. m3 Therefore, fx) x x 3 1 ab + )x + x 4

6 INTEGERS: 9 009) 644 and as a result, we get 1 bx x ) F x) x + a b)x x x 3 1 ab + )x + x 4. After simplifying the above expression we get the desired result F x) x 1 + ax x ) 1 ab + )x + x Binet s Formula for the Generalized Fibonacci Sequence and Identities Koshy refers to the Fibonacci numbers as one of the two shining stars in the vast array of integer sequences [18]. We may guess that one reason for this reference is the sheer quantity of interesting properties this sequence possesses. Further still, almost all of these properties can be derived from Binet s formula. A main objective of this paper is to demonstrate that many of the properties of the Fibonacci sequence can be stated and proven for a much larger class of sequences, namely the generalized Fibonacci sequence. Therefore, we will state and prove an extension of Binet s formula for the generalized Fibonacci sequences and so derive a number of mathematical properties including generalizations of Cassini s, Catalan s, and d Ocagne s identities for the ordinary Fibonacci sequence. Theorem 5. Generalized Binet s formula) The terms of the generalized Fibonacci sequence {q m } are given by ) a 1 ξm) α m β m q m ab) m α β where α ab+ a b +4ab, β ab a b +4ab, and ξm) : m m. Proof. First, note that α and β are roots of the quadratic equation and ξm) x abx ab 0 { 0 if m is even 1 if m is odd is the parity function. We have seen that the generating function for the sequence {q m } is given by see Theorem 1) F x) x 1 + ax x ) 1 ab + )x + x 4.

7 INTEGERS: 9 009) 645 Using the partial fraction decomposition, we rewrite F x) as F x) 1 [ ] aα + 1) αx aβ + 1) βx α β x α + 1) x β + 1) where α and β are as above. Since the Maclaurin series expansion of the function A Bz z C is given by A Bz z C BC n 1 z n+1 AC n 1 z n, n0 the generating function F x) can be expressed as [ ] 1 aα + 1) αx aβ + 1) βx α β x α + 1) x β + 1) n0 [ 1 ] βα + 1) m+1 + αβ + 1) m+1 α β α + 1) m+1 β + 1) m+1 x m+1 m0 [ + a ] β + 1)α + 1) m+1 α + 1)β + 1) m+1 α β α + 1) m+1 β + 1) m+1 x m. m0 We now simplify using the following properties of α and β. i) α + 1)β + 1) 1, ii) α + β ab, iii) α β ab, iv) α + 1 α ab, vii) αβ + 1) β. β v) β + 1, vi) βα + 1) α, ab Using the above identities, we get F x) m0 m0 ) m+1 1 βα m+ + αβ m+ x m+1 ab α β + m0 ) m 1 α m+1 β m+1 x m+1 + ab α β ) m+1 1 β + 1)α m+ α + 1)β m+ a x m ab α β m0 ) m 1 α m β m a x m. ab α β

8 INTEGERS: 9 009) 646 Combining the two sums, we get F x) ) m 1 a 1 ξm) α m β m ab α β xm m0 q m x m. Therefore, for all m 0, we have ) a 1 ξm) α m β m q m ab) m α β. m0 Note that when a b 1, q m αm β m α β, which is the original Binet formula for the Fibonacci numbers. Theorem 6. Cassini s identity) For any nonnegative integer n, we have a 1 ξn) b ξn) q n 1 q n+1 a ξn) b 1 ξn) q n a 1) n. Since Cassini s identity is a special case of Catalan s identity, which is stated below, it is enough to prove Catalan s identity. Theorem 7. Catalan s identity) For any two nonnegative integers n and r, with n r, we have a ξn r) b 1 ξn r) q n r q n+r a ξn) b 1 ξn) q n a ξr) b 1 ξr) 1) n+1 r q r. Proof. Using the extended Binet s formula, we get a ξn r) b 1 ξn r) q n r q n+r a ξn r) b 1 ξn r) a 1 ξn r) ab) n r ) ) a 1 ξn+r) α n r β n r α n+r β n+r ab) n+r α β α β a ξn r) b 1 ξn r) ) α n r β n r α n+r β n+r α β α β ab) n ξn r) and ) a α n αβ) n r α r + β r ) + β n ab) n 1 α β) ) a a ξn) b 1 ξn) qn a ξn) b 1 ξn) ξn) α n αβ) n + β n ab) n α β) ) a α n αβ) n + β n ab) n +ξn) 1 α β) ) a α n αβ) n + β n ab) n 1 α β).

9 INTEGERS: 9 009) 647 Therefore, a ξn r) b 1 ξn r) q n r q n+r a ξn) b 1 ξn) q n a ab) n 1 ) αβ) n αβ) n r α r + β r ) α β) ) a ab) n 1 αβ) n r αr α r β r + β r α β) ) a α ab) n 1 ab) n r r β r ) α β 1) n+1 r a ab) r 1 ab) r a ξr) q r 1) n+1 r a ξr) 1 ab) 1 ξr) qr 1) n+1 r a ξr) b 1 ξr) qr. Theorem 8. d Ocagne s identity) For any two nonnegative integers m and n with m n, we have a ξmn+m) b ξmn+n) q m q n+1 a ξmn+n) b ξmn+m) q m+1 q n 1) n a ξm n) q m n. Proof. First note that ξm + 1) + ξn) ξmn + n) ξm) + ξn + 1) ξmn + m) 1 ξm n) and ξm n) ξmn + m) + ξmn + n). Using the extended Binet s formula and the above identities, we obtain: a ξmn+m) b ξmn+n) q m q n+1 ) aab) n and ab) m n ξm n) α m+n+1 + β m+n+1 αβ) n βα m n + αβ m n ) α β) a ξmn+n) b ξmn+m) q m+1 q n ) aab) n ab) m n ξm n) α m+n+1 + β m+n+1 αβ) n α m n+1 + β m n+1 ) α β).

10 INTEGERS: 9 009) 648 Therefore, a ξmn+m) b 1 ξmn+m) q m q n+1 a ξmn n) b 1 ξmn n) q m+1 q n 1) n a ab) m n ξm n) ) α m n β m n ) α β 1) n aa ξm n) 1 q m n 1) n a ξm n) q m n. Theorem 9. Additional identities) i) For any two nonnegative integers m and n, a ξmn+m) b ξmn+n) q m q n+1 + a ξmn+n) b ξmn+m) q m 1 q n a ξm+n) q m+n. This identity is equivalent to the Convolution Property given by a ξkm) b ξkm+k) q m q k m+1 + a ξkm+k) b ξkm) q m 1 q k m a ξk) q k. ii) For any two nonnegative integers n and k with n k, a 1 ξn+k) b ξn+k)) qn+k+1 + a ξn k) b 1 ξn k)) qn k aq n+1 q k+1. iii) For any natural number n, and iv) If m n, then q m q n. n+1 k1 k1 aq k 1 q k q n a ) [q aq k 1 q k b n+1 1 ]. v) For any two natural numbers n and m, we have gcd q m, q n ) q gcdm,n). vi) For any nonnegative integer n, q n+ q n a 1 ξn) b ξn) q n+ and q n+ + q n a 1 ξn) b ξn) q n+ + q n. Consequently, ) a 1 ξn) b ξn) q n+ + qn q n+ ) a 1 ξn) b ξn) q n+ + qn). This identity produces Pythagorean triples involving generalized Fibonacci numbers.

11 INTEGERS: 9 009) 649 For any three nonnegative integers n, k, j with k j, a 1)n ξj)ξn+k) b 1)n ξj)ξn+k) q k q n+j a 1)n ξk)ξn+j) b 1)n ξk)ξn+j) q j q n+k 1) j q n q k j. Proof. We leave the proofs to the reader, since they are similar to the proof of the previous theorem. Theorem 10. Sums Involving Binomial Coefficients). For any nonnegative integer n we have and k0 k0 ) n a ξk) ab) k q k q n k ) n a ξk+1) ab) k+1 q k+1 aq n+1. k Proof. First note that a αk β k α β ab) k a ξk) q k for any nonnegative integer k. Therefore, k0 ) n a ξk) ab) k q k k n )a αk β k k α β k0 [ a n ) n α k α β k k0 a k0 α β [α + 1)n β + 1) n ] [ ) a α n ) β n ] α β ab ab a α n β n ) ab) n α β q n. ] n )β k k

12 INTEGERS: 9 009) 650 Also, k0 ) n a ξk+1) ab) k+1 q k+1 k n )a αk+1 β k+1 k α β k0 [ a ) n α α k β α β k k0 a k0 α β [αα + 1)n ββ + 1) n ] [ ) a α n ) β n ] α β α β ab ab a α n+1 β n+1 ) ab) n α β aq n+1. ] n )β k k Remark 11 The last two identities can be combined and generalized as follows. For any nonnegative integer r, one can easily verify either using the generating function together with the differential operator or the extended Binet Formula) that k0 ) n a ξk+r) ab) k +ξk)ξr) q k+r a ξr) q n+r. k 4. A Further Generalization, Convergence Properties, and an Open Problem Now, we may take the generalized Fibonacci sequence a bit further by allowing arbitrary initial conditions. So, consider the sequence {Q n }, where Q 0 C, Q 1 D, and { aqn 1 + Q Q n n, if n is even bq n 1 + Q n, if n is odd n ). The following theorem states a relationship between the terms of {Q n } and the terms of {q n }, the generalized Fibonacci sequence. In addition, the generating function for the {Q n } is given. First part of this theorem can be proven by induction. Theorem 1. Let the sequence {Q n } satisfy the above initial conditions and recurrence relation. Then Q n Dq n + C ) ξn) b q n 1 a

13 INTEGERS: 9 009) 651 for all n 1. Moreover, its generating function is given by Gx) n0 Q n x n C + Dx + ad abc C)x + bc D)x 3 1 ab + )x + x 4. The {q n }-identities we studied in this article can be extended to the sequence {Q n } with minor modifications. Remark 13 On Convergence Properties). For the classical Fibonacci sequence {F n }, which is {q m } with a b 1), it is well-known that the ratios of successive terms { } Fn+1 F n converge to the golden ratio, or golden mean, φ Consider the generalization obtained when a b. From Theorem, we get q m+1 1 q m a αm+1 β m+1 α m β m α 1 a 1 ) m+1 β α β α ) m. As a result, qm+1 q m converges to α a a+ a +4. Of course, when a 1, the quadratic irrational α a is called the golden mean. V. W. de Spinadel in [4] gives the names silver mean and bronze mean to the cases when a and a 3, respectively. Now, if a b, the ratios of successive terms do not converge since q m+1 q m a ξm) b 1 ξm) + q m 1 q m and { a ξm) b 1 ξm)} oscillates between a and b. Therefore, for most sequences in the family of generalized Fibonacci sequences, the ratios of successive terms do not converge. However, it is not hard to see that q m q m 1 α b, q m+1 q m α a, and q m+ q m α + 1. Remark 14 An Open Problem). Let a 1, a,..., a k be positive integers and define a sequence {q m } as follows. Set and for all m, q 0 0, q 1 1, q m a t q m 1 + q m where m t + 1 mod k) for some t {1,..., k}. When k, {q m } is the family of generalized Fibonacci sequences we studied in this paper. It remains open to find a closed form of the generating function and a Binet-like formula for {q m }, provided they exist. Acknowledgments We would like to thank Professor Doron Zeilberger of Rutgers University for his excellent comments. Also, we would like to thank the referees for their useful suggestions.

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