Applicable Analysis and Discrete Mathematics available online at Roman Witu la, Damian S lota

Transcript

1 Applicable Analysis Discrete Mathematics available online at Appl. Anal. Discrete Math , doi: /aadm w δ-ibonacci NUMBERS Roman Witu la, Damian S lota The scope of the paper is the definition discussion of the polynomial generalizations of the ibonacci numbers called here δ-ibonacci numbers. Many special identities interesting relations for these new numbers are presented. Also, different connections between δ-ibonacci numbers ibonacci Lucas numbers are proven in this paper. 1. INTRODUCTION Rabinowitz in [9] Grzymkowski Witu la in [4] independently discovered studied the following two identities: ξ + ξ 4 n n+1 + n ξ + ξ ξ 2 + ξ 3 n n+1 + n ξ 2 + ξ 3, where n, n N denotes the ibonacci numbers see, for example [7] ξ C, ξ 1, ξ 1. Corresponding to these identities, many classical identities relations for ibonacci Lucas numbers could be proved in a new elegant way see for example [12]. In this paper we generalize the identities to the following forms: δξ + ξ 4 n an δ + b n δξ + ξ δξ 2 + ξ 3 n an δ + b n δξ 2 + ξ Mathematics Subject Classification. 11B39, 11B83, 11A07, 39A10. Keywords Phrases. ibonacci numbers, Lucas numbers. 310

2 δ-ibonacci numbers 311 for δ C, n N. Simultaneously with these new identities, we introduce the polynomials a n δ, b n δ Z[δ], which by , can be treated as the generalization of the ibonacci numbers, therefore, which are called here the δ-ibonacci numbers. The scope of this paper is to investigate basic properties of these new kinds of numbers. or example, Binet s formulae for δ-ibonacci numbers formulae connecting a n δ b n δ, n N, with ibonacci Lucas numbers are presented. It is important to emphasize that our δ-ibonacci numbers belong to the family of the so called quasi-ibonacci numbers of k-th, δ-as orders for k 2N 1, δ C. These numbers are, by definition, the elements of the following sequences of polynomials: {a n,i δ} n N Z[δ], δ C, i 1, 2,..., 1 2 ϕk, where ϕ is the Euler s totient function. If k is an odd prime number, then the quasi-ibonacci numbers of k-th, δ-as order are determined by the following identities: δ ξ l + ξ k l k 3/2 n an,1 δ + a n,i+1 δ ξ i l + ξ ik l, i1 for l 1, 2,..., k 1/2, n N where ξ : exp2 π i/k. More information general definitions of the quasi-ibonacci numbers are given in papers [12, 14]. We note that δ-ibonacci numbers, are the simplest members of the family of the quasi-ibonacci numbers of k-th, δ-as order. Moreover, in paper [12] [4] the basic properties of the quasi-ibonacci numbers of the 7-th, δ-as 11- th, δ-as order, respectively, are presented. In paper [14], which is a continuation of paper [12], the applications of the quasi-ibonacci numbers of the 7-th, δ-as order to the decomposition of some polynomials of the third degree are studied. They make it possible to generate new Ramanujan-type trigonometric identities! Remark 1.1. Most identities relations discussed in the paper are proved by an immediate application of identities The use of Binet s formulae for δ- ibonacci numbers which were proved in Section 4 is restricted in this paper only to some cases. By doing so, we want to promote an alternative, more creative method of generating verifying the identities for δ-ibonacci numbers. Our paper is divided into seven sections. In the second section, the definitions notations are given. In the third section, the basic formulae properties of δ-ibonacci numbers are derived. In the fourth section of the paper, the Binet formulae for δ-ibonacci numbers are proven. The relationships between δ-ibonacci numbers for different values of δ s are studied in the fifth section of the paper. In the sixth section, the reduction formulae for indices are given. Some summation convolution type formulae for δ-ibonacci numbers are derived in the last section of the paper.

3 312 Roman Witu la, Damian S lota 2. DEINITIONS AND NOTATIONS Let ξ expi2π/. Let us start with the following basic result: Lemma 2.1. a Any two among the following numbers: 1, β : ξ + ξ 4 2 cos 2π 1, α : ξ 2 + ξ 3 2 cos π are linearly independent over Q. Moreover, we have [6, 7]: α + β α β 1, α 2 α + 1 β 2 β + 1. b Let f k Q[δ] g k Q[δ], k 1, 2. Then, for any a, b R linearly independent over Q, if f 1 δa + g 1 δb f 2 δa + g 2 δb, for δ Q then f 1 δ f 2 δ g 1 δ g 2 δ, for δ C. Now, let us set δξ + ξ 4 n an δ + b n δξ + ξ 4 for δ C, n N. Then by Lemma 2.1 the following recurrence relations hold true: a 0 δ 1, b 0 δ 0, 2.2 a n+1 δ a n δ + δb n δ, 2.3 b n+1 δ δa n δ + 1 δb n δ, or 2.4 [ an+1 δ b n+1 δ ] [ 1 δ δ 1 δ ] [ an δ b n δ ]. The elements of sequences {a n δ} {b n δ} will be called the δ-ibonacci numbers. Table 1: The most important cases for Axxxxxx see [10] δ 2 1 1/2 2/3 1 2 a n δ A01448 A n 2n+1 3 n 3n+1 n+1 n/2 b n δ 3n 2n 2 n 2n 3 n 3n n 1 1 n n/2

4 δ-ibonacci numbers 313 Remark 2.2. We note that, also by Lemma 2.1, the following identity holds: δ ξ 2 + ξ 3 n a nδ + b nδξ 2 + ξ 3. Hence, by 2.1 we get the identity: δ ξ + ξ 4 n δ ξ 2 + ξ 3 n 2 a nδ b nδ : A nδ. The new auxiliary sequence {A nδ} shall be used in many applications of numbers {a nδ} {b nδ}, especially for the decomposition of certain polynomials. Remark 2.3. Let us highlight that from definition the previously mentioned fact can be inferred that a nδ, b nδ Z[δ], n 0,1, 2,... Thus, in view of Lemma 2.1, it may be concluded that, if i, G i Q[x 0, x 1,..., x 2n+1], i 1, 2, 1 [ a0δ,a 1δ,...,a nδ, b 0δ, b 1δ,..., b nδ ] for δ Q, then + ξ + ξ 4 2 [ a0δ, a 1δ,..., a nδ,b 0δ, b 1δ,..., b nδ ] G 1 [ a0δ, a 1δ,..., a nδ,b 0δ, b 1δ,..., b nδ ] + ξ + ξ 4 G 2 [ a0δ, a 1δ,..., a nδ,b 0δ, b 1δ,..., b nδ ], i [ a0δ,...,b nδ ] G i [ a0δ,..., b nδ ], for every i 1,2 δ R. A similar fact holds when ξ + ξ 4 is replace by ξ 2 + ξ 3. The two facts discussed above referred to as reduction rules, constitute a principal technical trick used almost throughout our paper. Once again, it should be emphasized that this method of proving the identities for elements of recurrence sequences practically has not been used in the literature. It is our hope that this paper shall contribute to the popularization of this method, because of its clarity ease of operation in contrast to the generating functions based method Binet s formulae based method. 3. BASIC ORMULAE AND PROPERTIES Lemma 3.1. The following relations hold: a b n 0 0, 3.1 a n 2 n/2, b 2n 2 0, b 2n n b 2 δ b 2n δ. b We have for the proof see Corollary 6.3 : 3.2 n k+1 a n c k kn+1 k+1 n b k n kn. k+1 k b n+1 δ b n δ δ a n δ b n δ ;

5 314 Roman Witu la, Damian S lota 3.4 a n+k δ a k δa n δ + b k δb n δ; 3. b n+k δ b k δa n δ + [ a k δ b k δ ] b n δ; 3.6 a n+1 δ b n+1 δ [ a 2 δ b 2 δ ] a n 1 δ b n 1 δ + b 2 δb n 1 δ 3.7 [ 1 δ 2 + δ 2] a n 1 δ b n 1 δ + δ2 δb n 1 δ 3.8 δb n+1 δ 1 δa n+1 δ + δ 2 + δ 1a n δ. d The recurrence formulae for elements of {a n δ}-as {b n δ}-as: 3.9 a n+2 δ + δ 2a n+1 δ + 1 δ δ 2 a n δ b n+2 δ + δ 2b n+1 δ + 1 δ δ 2 b n δ 0. e or a, b, c, d C, c + dξ + ξ 4 0 d 2 c 2 + cd 0 : 3.11 a + bξ + ξ 4 bd + ad ac c + dξ + ξ 4 d 2 c 2 + cd + ad bc d 2 c 2 + cd ξ + ξ4. Proof. a We have So, by 2.4 we get: [ a2n+1 2 b 2n+1 2 [ ] [ a2n 2 b 2n 2 [ ] ] 2 [ ] 2n [ a1 2 b 1 2 [ n 1 a 2 2 n 1 b 2 2 ]. ] [ ] [ n 0 Since b 2 δ δ 2 δ, we have b 2 δ b 2n δ b 2n 0 b 2n 2 0. c Identity 3.3 follows from 2.3. or the proof of first, we note that: ] n 2 n, ] δ ξ + ξ 4 m+n am+n δ + b m+n δξ + ξ 4. Next, the following decomposition could be generated: δ ξ + ξ 4 m+n 1 + δ ξ + ξ 4 m 1 + δ ξ + ξ 4 n

6 δ-ibonacci numbers 31 a m δ + b m δξ + ξ 4 a n δ + b n δξ + ξ 4 a m δa n δ + b m δb n δ + a m δb n δ + a n δb m δ b m δb n δ ξ + ξ 4. Comparing 3.12 with 3.13 we see that by reduction rules, identities hold. Identity 3.7 is derived from Next, by again, we get: δb n+1 δ δ 2 a n δ + 1 δδb n δ δ 2 a n δ + 1 δ a n+1 δ a n δ which implies 3.8. d By we obtain: a n+2 δ a n+1 δ 1 δa n+1 δ + δ 2 + δ 1a n δ, which implies 3.9. On the other h, by 3.3 we obtain: b n+2 δ + δ 2b n+1 δ + 1 δ δ 2 b n δ b n+2 δ b n+1 δ + δ 1 b n+1 δ b n δ δ 2 b n δ δ a n+1 δ b n+1 δ + δ 1δ a n δ b n δ δ 2 b n δ δ [ a n+1 δ a n δ δb n δ + δa n δ + 1 δb n δ b n+1 δ ], which is equal to zero, by again. The δ-ibonacci numbers are described in an explicit form in the following: Theorem 3.2. The following decompositions hold k+1 k + k 1, k Z; 3.14 a n δ 3.1 b n δ k1 n k k 1 δ k n k 1 k 1 k δ k. see also.6.7 where alternative formulae are given. Proof. Both formulae follow from Corollary 3.3. We have: 3.16 A n δ where L k denotes the k-th Lucas number. n k δ k L k,

7 316 Roman Witu la, Damian S lota Remark 3.4. The following formulae could easily be derived: a nδ nb n 1δ, b nδ n a n 1δ b n 1δ, bnδ n1 δ δ2 n 1 a. nδ 2 anδ Hence, the following relations could be deduced: 3.17 a 2nδ > 0; 3.18 b 2δb 2nδ 0, but, b 2nδ 0 b 2δ 0 δ 0 δ 2; 3.19 a 2nδ > b 2nδ; 3.20 b 2n+1δ > b 2nδ for δ > b 2n+1δ < b 2nδ for δ < 0; 3.22 a n+1δ > a nδ > 0 for δ < 0; 3.23 δb 2n+1δ 0, but b 2n+1δ 0 δ 0; [ ] a 2n+1δ > 0 for δ, BINET ORMULAE OR δ-ibonacci NUMBERS The characteristic equation of recurrence formulae has the following decomposition: 4.1 X 2 + δ 2 X + 1 δ δ 2 X 1 + δ 1 X 1 + δ π π X 1 2 δ cos X δ cos. Hence, the following two identities follow: Theorem 4.1 Binet formulae for a n δ-as b n δ-as. We have: 4.2 a n δ + 10 n 2 δ + δ + 2 δ δ n 2 δ + δ 2 δ δ 4.3 b n δ 2 2 for every n 1, 2,.... n n

8 δ-ibonacci numbers 317 Corollary 4.2. Using formulae the following identities can be generated: a n δ + 2 b n δ n 2 δ + δ + 1, b n δ 1a n δ 1 n 2 δ δ. 2 The next identity is derived by multiplying : 4.6 b 2 n δ a 2 n δ + a n δb n δ 1 δ δ 2 n. urthermore from we get: 4.7 a 2 n δ + b 2 n δ a 2nδ, 4.8 a n δb n δ a 2n δ + 2 b 2n δ 1 δ δ 2 n. Corollary 4.3. We have: 4.9 a n k δa n+k δ a 2 n δ b 2 n δ b n kδb n+k δ bk 2 δ1 δ δ 2 n k the Catalan s-type formula, { δ 2 1 δ δ 2 n 1 for k 1 the Simson s-type formula, δ 2 δ 2 δ + 1 δ δ 2 n 2 for k 2. urthermore, the Tagiuri s-type formula [3] holds: a n+k δa n+l δ a n δa n+k+l δ b n δb n+k+l δ b n+k δb n+l δ Moreover, we note that the following expression: 1 δ δ 2 n b k δb l δ. a n 2 δa n 1 δa n+1 δa n+2 δ a 4 n δ is independent from an 2 δ iff δ 0 or δ 1. or δ 1 we get the Gelin-Cesáro identity see [3,, 8]: n 2 n 1 n+1 n+2 4 n 1. Proof. Let us set x n δ r u n + s v n, n N. Then, we obtain: x 2 n δ x n kδx n+k δ r s u k v k 2 u v n 1. Hence, using the formula 4.9 follows.

9 318 Roman Witu la, Damian S lota Corollary 4.4. We have: 4.10 a 2 n δ 3 a 2n δ + b 2n δ δ δ 2 n, 4.11 b 2 n δ 2 a 2n δ b 2n δ 2 1 δ δ 2 n, 4.12 a 3 n δ 2 a 3n δ + b 3n δ δ δ 2 n a n δ, 4.13 bn 3 δ b 3n δ 3 1 δ δ 2 n b n δ, k 4.14 k bn 2k 2k 1 l l 1 δ δ 2 ln 2a 2k ln b 2k ln. l0 Remark 4.. rom 3.14 Corollaries the following identities for every 0 m 2n can be derived: m n k n m k k 1 m k 1 m n k m n k m n k m n k n m k k m k n L m k k L m k n m k k 1 L m k n m k k L m k 2 n L m m m m! 2 n L m m 2 1m m! 2 n L m m + 2 1m m! 2 n m 1 + 1m m m! 2 n m. m d m dδ m 1 δ δ2 n δ0, d m dδ m 1 δ δ2 n δ0, d m dδ m 1 δ δ2 n δ0, d m dδ m 1 δ δ2 n δ0, Also, see papers [1, 2] for interesting recurrence relations for the coefficients of polynomials 1 + δ + δ 2 n, n N.. THE RELATIONSHIP BETWEEN δ-ibonacci NUMBERS OR DIERENT VALUES O δ s Let us start with the following fundamental formulae, which make it possible to define numbers a n δ b n δ by a given a n b n with δ. Theorem.1. The following identities hold: δ.1 ζ n a n ζ δ.2 ζ n b n ζ n k ζ 1 n k a k δ n k ζ 1 n k b k δ

10 δ-ibonacci numbers 319 for ζ 1 ζ 0 δ 0. Proof. By 2.1 we have for ζ 0:.3 ζ + δξ + ξ 4 n ζ n 1 + δ n δ δ ζ ξ + ξ4 ζ n a n + ζ n b n ξ + ξ 4. ζ ζ On the other h, we obtain: ζ + δξ + ξ 4 n ζ δξ + ξ 4 n.4 n k ζ 1 n k 1 + δξ + ξ 4 k n k ζ 1 n k a k δ + n n ζ 1 k n k b k δ ξ + ξ 4. Using.3,.4 the reduction rules identities.1.2 follow. Corollary.2. or ζ 2, δ 2 we obtain a new identity as it seems :. 2 n 2n r Corollary.3. or δ 1 ζ 1/η we get:.6 a n η.7 b n η n k 3k r, for every r Z. n k 1 η n k η k k+1 n k 1 η n k η k k. Hence, for η 1, the following two known formulae follow: 2n 1 2 n n n k 1 2 kk+1, 2n 2 n n n k 1 2 kk. Remark.4. Comparing formulae.6.7 with formulae respectively, two new interesting identities can be observed:.8.9 n k 1 η n k η k k+1 n k 1 η n k η k k n k η k k 1, n k η k k,

11 320 Roman Witu la, Damian S lota i.e.,.10 0 n [1 η n k + 1 k] η k k k. Theorem.. The following four identities hold: 1.11 a n δ n+1 + b n δ n 1 + δ n a n 1 + δ.12 a n δ n + b n δ n δ n b n δ.13 n+1 a n δ n+2 b n δ 1 2 δ n a n 1 δ 1 2δ 1 δ.14 n a n δ n+1 b n δ 1 2 δ n b n. 1 2δ,, Proof. Immediately from the identity δ 1: ξ + ξ δξ + ξ δ δ ξ + ξ4 we obtain: 1 + ξ + ξ 4 n 1 + δξ + ξ 4 n 1 + δ n n, 1 + δ ξ + ξ4 i.e., n+1 + n ξ + ξ 4 a n δ + b n δξ + ξ δ n 1 1 a n + b n 1 + δ 1 + δ a n δ n+1 + b n δ n + a n δ n b n δ n ξ + ξ δ n a n δ ξ + ξ 4, δ n b n δ, by applied the reduction rules identities follow. The following identity can be obtained in a similar way: ξ + ξ δ ξ 2 + ξ δ ξ 2 + ξ 3 + ξ + ξ 4 δ ξ + ξ 4, 1 2 δ + 1 δξ + ξ δ δ 1 2δ ξ + ξ4.

12 δ-ibonacci numbers 321 Raising both sides of the above identity to the n-th power applying 2.1, we obtain: n+1 + n ξ + ξ 4 a n δ + b n δξ 2 + ξ 3 1 δ 1 δ 1 2δ n a n + b n ξ + ξ 4 1 2δ 1 2δ which, after some calculations, yields Theorem.6. We have:.17 a n 1 ε 2 1 ε n 1 1 a n εa n + b n εb n, 1 ε 1 ε.18 b n 1 ε 2 1 ε n a n εb n +a n b n ε b n εb n, 1 ε 1 ε 1 ε.19 1 ε ε δ n ε δ a n a n δa n ε a n δb n ε b n δb n ε, ε δ + ε ε ε δ n b n ε δ a n εb n δ a n δb n ε. ε δ + ε 1 Proof. Immediately from identity: ε 2 ξ + ξ ε ξ + ξ 4 1 ε + ξ + ξ 4, we obtain: a n 1 ε 2 + b n 1 ε 2 ξ + ξ 4 1 ε n a n ε + b n εξ + ξ a n + b n ξ + ξ 4, 1 ε 1 ε which, after easy calculations implies Next, we note that 1 + µ ξ + ξ 4 n 1 + δ ξ 2 + ξ 3 n 1 δ µ δ + µ δξ + ξ 4 n 1 δ µ δ n a n On the other h, we get: 1 + µ ξ + ξ 4 n 1 + δ ξ 2 + ξ 3 n µ δ µ δ + b n ξ + ξ 4. 1 δ µ δ 1 δ µ δ a n µ + b n µξ + ξ 4 n an δ + b n δξ 2 + ξ 3 n a n µa n δ b n µb n δ a n µb n δ + a n δb n µ a n µb n δ ξ + ξ 4. Comparing both of these decompositions by the reduction rules formulae can be deduced.

13 322 Roman Witu la, Damian S lota 6. REDUCTION ORMULAE OR INDICES Theorem 6.1 The following reduction formulae for indices hold: a ormulae from Section 3; b a mδ + b m δ a m δ a n δ b n δ 6.1 a m n δ a nδ + b n δ a n δ a n δ b n δ a mδ b m δ a n δ b n δ 6.2 b m n δ a nδ + b n δ a n δ a n δ b n δ ; c 6.3 a m n δ a n bm δ mδa n, a m δ 6.4 b m n δ a n bm δ mδb n, a m δ bmδ bn 6. a m n p δ a np bm δ a mδ m δap n a p, a m δ bmδ a n a mδ bmδ bn 6.6 b m n p δ a np bm δ a mδ m δap n b p. a m δ bmδ a n a mδ Proof. b We have: a m n δ + b m n δξ + ξ δ ξ + ξ 4 m n 1 + δ ξ + ξ 4 m 1 + δ ξ + ξ4 n a mδ + b m δξ + ξ 4 a n δ + b n δξ + ξ 4 formulae b from Lemma 3.1 e follows. c We have: a m n δ + b m n δξ + ξ δ ξ + ξ 4 m n am δ + b m δξ + ξ 4 n a n m δ bm δ bm δ a n + b n ξ + ξ 4, a m δ a m δ

14 δ-ibonacci numbers 323 which yields identities Similarly, identities can be deduced. Corollary 6.2. The following identity holds: 6.7 an δ b n δ a n+k δ b n δb n+k δ 1 δ δ 2 n a k δ. Proof. The identity follows from 3.4, Corollary 6.3. rom we get, for δ 1 : 6.8 mn+1 n m+1 a n m m mn m n m b n m mn mn m bn 6.10 mnp+1 np m m+1 ap n a p m+1 m bn 6.11 mnp m np m ap n b p m+1, next, for δ 1 : mn 1 n 2m 1 a n mn n 2m 1 b n etc. 2m 2m 1 2m 2m 1 m+1, m a n m+1 m+1, m a n m mn mn n k k 1 m k n k m+1, n k 1 k 1 k k m n k m+1, n k k 1 k 2m n k 2m 1, n k k k 2m n k 2m 1, Remark 6.4. All formulae from Corollary 6.3 are simultaneously generalizations variations of the known Church Bicknell s identities [7, p. 238]. Remark 6.. Professor W. Webb asked on the 13th International Conference on ibonacci Numbers Their Applications in Patras july 2008 on closed formulae for the sums N k r, k1 where r is a fixed positive integer. Admittedly, formulae 6.9, 6,11, etc., do not imply such closed formulae, but allow for the reduction the problem to the sum of the powers of the ibonacci numbers with indices not greater than N. or example, we have: 6.14 N k1 k N k1 k k k b k k+1 N k k1 l0 k l 1 l 1 l l k k l k+1,

15 324 Roman Witu la, Damian S lota 6.1 N k1 k N k bk k k2 a k k k b k+1 k k+1 k. a k k+1 k1 Remark 6.6. rom 1.1 we obtain: 1 + ξ + ξ 4 n ξ + ξ 4 n 1 L n+1 + L n ξ + ξ 4, i.e., ξ + ξ ξ + ξ 4 n L n+1 + L n ξ + ξ 4. Hence, by we get: a k 2 + b k 2 ξ + ξ 4 kn+1 + kn ξ + ξ 4 L k n+1 a k Ln L n+1 which, after some manipulations, gives us two new formulae: a k 2 kn+1 + b k 2 kn L k Ln n+1a k L n+1 Ln ξ + b k + ξ 4, L n+1 a k 2 kn + b k 2 kn+1 kn ak 2 kn + b k 2 kn 1 L k Ln n+1b k. L n+1 inally, taking into account equalities 3.1 we obtain: 6.16 k 2kn+1 L 2k Ln n+1 a 2k L n k L 2k+1n+1 L 2k+1 Ln n+1 a 2k+1 L n k 2kn L 2k Ln n+1 b 2k L n+1 2k 2 k r r r 1 Ln L 2k r n+1, r0 2k+1 r0 2k r0 2 k + 1 r r 1 Ln L 2k+1 r n+1, r 1 r 1 2 k r rl r n L 2k r n+1, 2k k L 2kn L 2k+1 Ln n+1 b 2k+1 1 r 1 2 k + 1 L n+1 r r0 rl r n L 2k+1 r n SOME SUMMATION AND CONVOLUTION TYPE ORMULAE Theorem 7.1. a Let µ, δ C N N. Then, we have: 7.1 µ δ + µ 1 µ 1 µ 1 µ δ µ n a n δ

16 δ-ibonacci numbers 32 δ 2 + δ 1µ N+1 a δ + µ N a N δ + µ µ δ µ δ + µ 1 µ 1 µ 1 µ δ µ n b n δ µn a N δ 1 µ N b N δ µ 1 µ δ. In the sequel, we obtain: 7.3 δ 7.4 δ a n δ a N δ + b N δ 1 b n δ a N δ b or every r, N N, we have: a rnδ + b rn δ 1 a rn δ 1 a r δ 1 b r δ a rn δ a rδ + b r δ 1 a r δ 1 a r δ 1 b r δ 7.6 b rn δ a rnδ 1 b rn δ a r δ 1 b r δ a rδ + b r δ 1 a r δ 1 a r δ 1 b r δ. Proof. We have: δ ξ + ξ 4 rn δ ξ + ξ 4 rn δ ξ + ξ 4 r 1 a rnδ b rn δξ + ξ 4 1 a r δ b r δξ + ξ 4 b rδ b rn δ + a rn δ 1 a r δ 1 a rn δ 1 br 2δ 1 a r δ 2 + br δ a r δ 1 + b rδ a rn δ 1 b rn δ a r δ 1 br 2 δ 1 a r δ 2 + br δ a r δ 1 ξ + ξ

17 326 Roman Witu la, Damian S lota On the other h, we have: 1 + δ ξ + ξ 4 r n arn δ + b rn δξ + ξ 4 a rn δ + ξ + ξ 4 b rn δ, which, by 7.7, implies Theorem 7.2. Let µ, δ, ω C k, l, N N. Then, we have: a 2 N 1 N N an a N 2 µ + δ µa n N n δ + b n µb N n δ, 2 N 1 N N an b N 2 µ + δ µb n N n δ + a N n δb n µ b n µb N n δ ; N b δ a n δa N n δ N + 1δ 3 a N δ + b N δ + 2 b N+1 δ, N δ b n δb N n δ N + 1δ 2 a N δ b N δ 2 b N+1 δ, N δ a n δb N n δ N + 1δ a N δ + 2 b N δ b N+1 δ. c In addition, if ω + a l δ 0, N N ω n N n a n l+k δ ω + a l δ N bl δ a k δa N ω + a l δ N N ω n N n b n l+k δ ω + a l δ N + b k δ a N bl δ + b k δb N ω + a l δ [ a k δb N bl δ ω + a l δ b N bl δ ω + a l δ, ] bl δ. ω + a l δ Proof. c irst, we note that Sδ; ω : 1 + δ ξ + ξ 4 k ω δ ξ + ξ 4 l N a k δ + b k δξ + ξ 4 ω + a l δ + b l δξ + ξ 4 N

18 δ-ibonacci numbers 327 ω + a l δ [ N ak δ + b k δξ + ξ 4 bl δ a N ω + a l δ ] bl δ + b N ξ + ξ 4 ω + a l δ ω + a l δ [ ] N bl δ bl δ a k δa N + b k δb N ω + a l δ ω + a l δ + ω + a l δ [ N bl δ a k δb N ω + a l δ ] bl δ bl δ + b k δ a N b N ξ + ξ 4. ω + a l δ ω + a l δ On the other h, by an application of the binomial formula to Sδ; ω, we get Sδ; ω N N N n ω N n 1 + δ ξ + ξ 4 n l+k N n ω N n a n l+k δ + ξ + ξ 4 N N n ω N n b n l+k δ. Comparing two decompositions of Sδ; ω formulae c follows. Remark 7.3. The formulae c of Lemma 7.2 are, in some sense, the generalizations of the following variations of the de Vries identities see [11], where only the case of ω l 1 is discussed: N N ω N n n nl+i α i ω + α l N β i ω + β l N, N N n ω N n L nl+i α i ω + α l N + β i ω + β l N, where α β in Lemma 2.1 are defined. The proof of can be obtained by an immediate application of the Binet s formula for the ibonacci Lucas numbers. Moreover, from both for δ 1 from we deduce the formulae: l+1 ω 1 + l+1 ω N i a N N L i a N which implies the following relations: l + i 1 b N ω + l+1 ω N α i ω + α l N β i ω + β l N, l ω + l+1 l ω + l+1 + L i 1 b N l ω + l+1 ω N α i ω + α l N + β i ω + β l N,

19 328 Roman Witu la, Damian S lota i ω + l+1 N a N l ω + l+1 i 1 Li 1 α i ω + α ln i 1 ω + l+1 N b N l ω + l+1 i Li α i ω + α ln + i 1 + Li 1 β i ω + β ln, i + Li β i ω + β ln. At the end of this section, two convolution identities will be presented. Theorem 7.4. We have: n k δ n k b k δ n k n k δ n k b k δ n k+1 n k δ n k a k δ n k 1, n k δ n k a k δ n k. inal remarks. As noticed by one of the reviewers, identities have a corresponding form for a general second order recurrence u 0 0, u 1 1, u n+1 Au n +B u n 1, n N. Then we have λ n u n λ+b u n 1 λ n u n λ+b un 1, where λ A + A 2 + 4B /2 λ A A 2 + 4B /2. Acknowledgments. The authors wish to express their gratitude to the Reviewers for several helpful comments concerning the first version of the paper. REERENCES 1. L. Carlitz: Some combinatorial identities. Univ. Beograd. Publ. Elektrotehn. ak. Ser. Mat. iz., , D. M. Cvetković, S. K. Simić: On enumeration of certain types of sequences. Univ. Beograd. Publ. Elektrotehn. ak. Ser. Mat. iz., , S. airgrieve, H. W. Gould: Product difference ibonacci identities of Simson, Gelin-Cesaro, Tagiuri generalizations. ibonacci Quart., R. Grzymkowski, R. Witu la: Calculus Methods in Algebra, Part One. WPKJS, Gliwice, 2000 in Polish.. A.. Horadam, A. G. Shannon: Generalization of identities of Catalan others. Port. Math., R. Knott: ibonacci golden ratio formulae T. Koshy: ibonacci Lucas Numbers with Application. Wiley, New York, J. Morgado: Note on some results of A.. Horadam A. G. Shannon concerning a Catalan s identity on ibonacci numbers. Port. Math.,

20 δ-ibonacci numbers S. Rabinowitz: Algorithmic manipulation of ibonacci identities. In: Applications of ibonacci Numbers, Vol. 6, eds. G. E. Bergum et al. Kluwer, 1996, pp N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences. njas/sequences/, P. C. G. de Vries: Problem 44, Nieuw Arch. Wiskd. 12. R. Witu la, D. S lota, A. Warzyński: Quasi-ibonacci numbers of seventh order. J. Integer Seq., , Article , R. Witu la, D. S lota: New Ramanujan-type formulas quasi-ibonacci numbers of order 7. J. Integer Seq., , Article R. Witu la, D. S lota: Quasi-ibonacci numbers of order 11. J. Integer Seq., , Article 07.8., Institute of Mathematics, Received September 14, 2008 Silesian University of Technology, Revised May 1, 2009 Kaszubska 23, Gliwice, Pol E mail: d.slota@polsl.pl