Roman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]:

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1 Novi Sad J. Math. Vol. 43 No δ-fibonacci NUMBERS PART II Roman Witu la 1 Abstract. This is a continuation of paper [6]. We study fundamental properties applications of the so called δ-fibonacci numbers a n δ b n δ. For these numbers many special identities interesting relations can be generated. Also the formulas connecting the numbers a n δ b n δ with Fibonacci Lucas numbers are presented. Moreover some polynomials generated by a n δ b n δ are discussed. AMS Mathematics Subject Classification 010: 11B39 11B83 Key words phrases: Fibonacci number Lucas number 1. Introduction Let ξ expiπ/. Then the following formulas hold true [6]: δξ + ξ 4 n an δ + b n δξ + ξ δξ + ξ 3 n an δ + b n δξ + ξ 3 for δ C n N. Hence the below recurrent relations can easily be generated [6]: a 0 δ a 1 δ 1 b 0 δ 0 b 1 δ δ a n+1 δ a n δ + δb n δ b n+1 δ δa n δ + 1 δb n δ. We note that a n δ b n δ Z[δ] for every n N {0}. Both a n δ b n δ for n N are called the δ Fibonacci numbers. It should be emphasized that these numbers are the simplest form of more general classes of numbers introduced recently in the papers [ ]. Additionally in paper [6] the following Binet s formulas for a n δ b n δ are presented 1. a n δ + n δ + δ + n δ δ n n δ + δ δ δ 1.6 b n δ 1 Institute of Mathematics Silesian University of Technology Kaszubska 3 Gliwice Pol roman.witula@polsl.pl

2 10 Roman Witu la for every n 1... as well as the formulas connecting a n δ b n δ with Fibonacci Lucas numbers denoted by F n L n respectively: a n δ b n δ n n k n n k F k 1 δ k F k 1 k 1 δ k. We note that by the following equalities are true see also Remark.1: 1.9 a n 1 F n+1 a n 1 F n 1 a n F 3n 1 b n 1 F n b n 1 F n b n F 3n. The next lemma consists of some other basic technical facts which will turn out to be useful in the current paper. Lemma a We have β : ξ + ξ 4 cos π 1 α : ξ + ξ 3 cos π +1. b Any two of the numbers 1 α β are linearly independent over Q. Moreover we have α + β α β 1 α α + 1 β β + 1. c Let f k Q[δ] g k Q[δ] k 1. Then for any a b R linearly independent over Q if f 1 δa + g 1 δb f δa + g δb for δ Q then f 1 δ f δ g 1 δ g δ for δ C.. Reduction formulas This section concerns the main reduction formulas for the indices of δ Fibonacci numbers. It is a supplement of Section 6 from the paper [6]. At the beginning we get 1 + δ ξ + ξ 4 kn ak δ + b k δ ξ + ξ 4 n.1 a n kδ 1 + b kδ a k δ ξ + ξ4 n a n bk δ bk δ kδ a n + b n ξ + ξ 4. a k δ a k δ

3 δ-fibonacci numbers part II 11 On the other h we have. 1 + δ ξ + ξ 4 kn akn δ + b kn δ ξ + ξ 4. By comparing.1 with. using Lemma 1.1 b we obtain the following basic reduction identities formulas from [6]:.3 a kn δ a n bk δ kδ a n a k δ.4 b kn δ a n bk δ kδ b n. a k δ It appears that the above formulas carry important reduction properties which are especially noticeable in the background of the following summation formulas:..6 N N a k n+1δ Hence for δ 1 we have.7.8 N N b k n+1δ a kn k δ a k n a kn k δ b k n bk δ a k δ bk δ. a k δ N N Fk F k n+1 +1 Fk+1 kn a k n F k+1 N N Fk F k n+1 Fk+1 kn b k n F k+1 next for n 1 we get.9.10 N N F k +1 Fk+1 k Fk a k F k+1 N N F k Fk+1 k Fk b k. F k+1 Another example of using formulas.3.4 is the following F 36 F37 F36 b F37 a F36 b b F 36 + F 3 F 6 7 b 6 F6 F 7 + F 8 a F7 F 8 F 36 a F36

4 1 Roman Witu la F6 b 6 a F6 3 F 7 F 7 b b3 F 6 F 7 a 3 F6 F 7. We note that by we get F36 F36 a a 4 + F 36 F36 b 4 a [ b a F36 F 36 F a 36 + F 36 b F36 b 1 F36 F36 F 36 a 4 + F 3 b 4 [ F 36 a F36 a b F 36 a F36 + F 3 F 36 b b F 36 F a 36 b F 36 Remark.1. We note that the relations Fk+1a n Fk n F kn+1 F n Fk F k+1b n k+1 F k+1 hold for every k Z \ { 1} n N. i.e. For example we get F n F 3 b n F 3n F 1 n b n 1 n 1 F 3n b n F 3n ] a F36 F kn ]. similarly a n +F 3n 1. Corollary.. If δ C k N a k δ b k δ then.11.1 For example: a n kδ F n+1 a kn δ a n kδ F n b kn δ. a δ b δ δ e±iπ/4.

5 δ-fibonacci numbers part II 13 Hence we get a n e±iπ/4 F n+1 1 ± 1 i n b n e±iπ/4 F n 1 ± 1 i n for every n N. Moreover if δ C k N a k δ b k δ then lim dist α a k δ n a kn δ 0 n n where the symbol a kn δ denotes the set of all complex roots of n th order of a kn δ for every n N. If additionally δ R then n+1 lim a kn+1 δ α a k δ n where the symbol n+1 a kn+1 δ denotes now the only real root of a kn+1 δ. At the end of this section we present one more result of the reductive nature. Theorem.3. We have.13 a k an δ + b n δ ξ + ξ 4 + b k an δ + b n δ ξ + ξ 4 ξ + ξ 4 Proof. We have 1 s k! r! s! δ r s a r+sn+r δ + b r+sn+r δ ξ + ξ 4. rs 0 r+s k δ ξ + ξ 4 n k ξ + ξ 4 a k an δ + b n δ ξ + ξ 4 + b k an δ + b n δ ξ + ξ 4 ξ + ξ 4. On the other h we get δ ξ + ξ 4 n k ξ + ξ 4 1\δ 1 + δ ξ + ξ 4 n+1 1\δ 1 + δ ξ + ξ 4 n k Sums of some series 1 s k! r! s! δ r s 1 + δ ξ + ξ 4 r+sn+r. rs 0 r+s k According to formula 1.1 two following formulas can be easily deduced 3.1 n1 an δ + b n δ cos π 1 δ cos π 1

6 14 Roman Witu la whenever 1 + δ cos π\ > 1 i.e. δ + + 1\ > + 1\ 3. n1 an δ + b n δ cos π δ cos π 1 + δ cos π 1 whenever δ + + 1\ < + 1\. Next from equation 1. we get 3.3 n1 an δ b n δ cos π 1 δ cos π 1 whenever 1 δ cosπ\ > 1 i.e. δ 1\ > 1\ 3.4 n1 an δ b n δ cos π 1 δ cos π δ cos π whenever δ 1\ < 1\. Note that all the above formulas can be easily presented in a more general form. We think that the case δ 1 for Fibonacci numbers is the most representative the appropriate formulas are listed below: a b c 3. Fkn+1 + F kn cos π s n0 F sk+1 + F sk cos π F sk+1 + F sk cos π for any k s N 1 cos π sk cos π for any k N s R sk + ; Fkn+1 + F kn cos π s Fln+1 + F ln cos π r n0 F sk+lr+1 + F sk+lr cos π F sk+lr+1 + F sk+lr cos π for any s r N 1 cos π sk+lr for any k l N cos π sk+lr 1 s r R + ; 3.7 Fkn+1 F kn cos π s 1 Fsk+1 + F sk cos π 1 n0 for every s N;

7 δ-fibonacci numbers part II 1 d 3.8 n0 1 kn F kn cos π F s kn ks F ks+1 F ks cos π 1 for any k s N 1 1 k F k+1 F k cos π s 1 1 cos π ks 1 for k N s R + ; e kn F kn+1 F kn cos π s 1 ln F ln+1 F ln cos π r n0 1 1 ks+lr F sk+lr+1 F sk+lr cos π 1 for any k l s r N 1 1 k F k+1 F k cos π s 1 l F l+1 F l cos π r 1 1 cos π ks+lr 1 for any k l N s r R+. 4. Two-parameter convolution formulas Results presented in the current paper refer to Theorems..6 from [6]. In particular the identities generalize identities.19.0 from the paper [6]. Theorem 4.1. We have 4.1 n an k δ b k ε a k ε b n k δ n/ n+1 n k 4 ε δ n k 1 ε δ k+1 k n ak ε a n k δ b k ε b n k δ a k ε b n k δ a n k δ b k ε n/ n+1 n k 4 ε δ n k ε δ k k

8 16 Roman Witu la n an δ b n ε a n ε b n δ 1 ε δ n δ ε ε δ a n k b k a k b n k 1 ε δ 1 ε δ 1 ε δ 1 ε δ n/ n k ε + δ n k 1 ε δ k+1 k ε δ 1 ε δ n an ε a n δ b n ε b n δ a n ε b n δ a n δ b n ε 1 ε δ n ε δ ε δ a k a n k b k b n k 1 ε δ 1 ε δ 1 ε δ 1 ε δ ε δ δ ε a n b n a n b n 1 ε δ 1 ε δ 1 ε δ 1 ε δ n/ n k ε + δ n k ε δ k k 1 ε δ 1 ε δ for every ε δ C n N. Proof. Let ξ exp π i/. We have + εξ + ξ 4 + δ ξ + ξ 3 n 1 + ε ξ + ξ δ ξ + ξ 3 n n n 1 + ε ξ + ξ 4 k 1 + δ ξ + ξ 3 n k k n n ak ε + b k ε ξ + ξ 4 a n kδ + b n kδ ξ + ξ 3. k By equalities 1.10 we obtain ak ε + b k εξ + ξ 4 a n k δ + b n k δ ξ + ξ 3 a k ε + b k ε 1 an k δ b n k δ +1 a k ε a n k δ b k ε b n k δ 1 ak ε b n k δ + a n k δ b k ε Furthermore we get + an k δ b k ε a k ε b n k δ. + ε ξ + ξ 4 + δ ξ + ξ 3 n 1 ε + δ + n/ n k a n k b k + n/ k ε δ n k n k + 1 a n k 1 b k+1

9 δ-fibonacci numbers part II 17 where a : 4 ε δ b : ε δ. Moreover we have 1 + ε ξ + ξ δ ξ + ξ 3 1 ε δ + ε ξ + ξ 4 + δ ξ + ξ 3 1 ε δ 1 + εξ + ξ 4 n 1 + δ ξ + ξ 3 n + ε 1 ε δ ξ + ξ4 + δ 1 ε δ ξ + ξ 3 a n ε + b n ε ξ + ξ 4 a n δ + b n δ ξ + ξ 3 a n ε a n δ b n ε b n δ + a n ε b n δ ξ + ξ 3 + a n δ b n ε ξ + ξ 4 a n ε a n δ b n ε b n δ 1 an ε b n δ + a n δ b n ε + an δ b n ε a n ε b n δ.. Inner products of δ-fibonacci vectors The aim of this section is to determine the compact analytical description of the following three sums: S N δ : S 1N δ : a k δ b k δ a kδ S 3N δ : We achieve the assumed goal in two independent ways. The first way b kδ. Let us start with the identity written below see formula.8 in [6]: 1 δ δ k 1 + δ ξ + ξ 4 k 1 + δ ξ + ξ 3 k.1 Hence we obtain. a k δ + b k δ ξ + ξ 4 a k δ + b k δ ξ + ξ 3 a kδ a k δ b k δ b kδ. S N δ S 1N δ S 3N δ 1 δ δ k 1 1 δ δ N 1 1 δ δ 1 δ + δ 1 1 δ δ N N 1 k k δ + δ k.

10 18 Roman Witu la Remark.1. If 1 δ δ 1 i.e. δ 0 or δ 1 then we set S N δ S 1N δ S 3N δ : On the other h we have H lim δ 0 or δ 1 lim δ 0 or δ δ ξ + ξ 4 k ak δ + b k δ ξ + ξ 4 1 δ + δ 1 1 δ δ N N 1 δ δ N. a kδ + b kδ + a k δ b k δ b kδ ξ + ξ 4 which implies the identity.3 S N δ + S 3N δ + ξ + ξ 4 S 1N δ S 3N δ 1 + δ ξ + ξ 4 k δ ξ + ξ 4 N δ ξ + ξ 4 1 a N δ b N δξ + ξ 4 1 a δ b δξ + ξ 4 b δ 1 a N δ 1 a δ a N δ + b N δ a N δ b N δ 1 a δ b δ b δ 1 a δ + b δ 1 a δ b δ ξ + ξ 4. Hence we obtain two new identities b δ 1 a N δ 1 a δ a N δ + b N δ 1 S N δ + S 3N δ b δ 1 a δ + b δ 1 a δ b δ i.e. δ 0 1 ± :.4 δ δ +1 S N δ+s 3N δ 1 δ a N δ 1+ δ b N δ S 1N δ S 3N δ i.e. δ 0 1 ± : 1 a N δ b N δ 1 a δ b δ b δ 1 a δ + b δ 1 a δ b δ. δ δ + 1 S 1N δ S 3N δ δ a N δ 1 δ b N δ.

11 δ-fibonacci numbers part II 19 Finally from the identities..4. we get.6 S N δ S N δ S 1N δ S 3N δ + 3 S N δ + S 3N δ + S 1N δ S 3N δ 1 1 δ δ δ + δ N δan δ δ + 1 δ δ b N δ.7 S 3N δ 1 δ δ δ + δ N δan δ δ + 1 δ δ b N δ.8 S 1N δ 1 δ + δ 1 δ δ N δan δ δ + 1 δ δ b N δ. For example from.6 for N we obtain.9 δ δ + δ + 34 δ δ 8 7 δa 4 δ δ b 4 δ. Corollary.. Using 1.9 we get see also [1 ]: S 3N 1 S N 1 S 1N 1 S 3N 1 S N+1 1 S 1N+1 1 F k 1 N + 3 F N F N+1 L + 1 N ; N Fk+1 F N+1 + F N 1 N F N+ + F N 1 N L N+1 1 N ; F k F k+1 L N N ; F k 1 N + F 4N ; N Fk 1 N + + F 4N ; N F k 1 F k N 1 + F 4N+1.

12 0 Roman Witu la Moreover from.1 the following formula can be derived S 1N+1 1 N L k F k F k N L k F k N F k. Hence by.13.1 we get i.e N + F 4N+1.16 N L k F k F 4N+ N 1 N L k F k L 4N+ 3. The second way First we prove in two ways two auxiliary formulas:.17 δ.18 δ b k δ a N δ 1 a k δ a N δ + b N δ 1. Proof. Immediately from 1.3 we derive δ b k δ Next from 1.4 we obtain δ ak+1 δ a k δ a N δ a 0 δ a N δ 1. a k δ δ Hence by 1.3 again we get δ b k+1 δ + δ 1 δ b k δ a N+1 δ 1 + δ 1 a N δ 1 a N+1 δ + δ 1 a N δ δ. a k δ a N δ + b N δ 1.

13 δ-fibonacci numbers part II 1 Now by the Binet s formulas we obtain N a k δ δ δ + δ δ+ δ N 1 δ+ δ 1 δ + 1 δ δ δ δ N 1 δ δ 1 1 δ δ 3 δ + 1 δ + N+1 δ 1 δ δ 3 δ + 1 δ + δ N+1 1 δ + 1 δ 1 δδ + δ 4 δ δ + 1 δ δ δ δ + 1 δ δ δ 1 a N+ δ + δ b N+ δ δ 3 δ + δ + i.e. the following identity holds.19 δ δ + δ 4 Next from 1.3 it follows that δ b k δ a k δ δ 1 a N δ 1 + δ b N δ. a k+1 δ a k δ Hence by we obtain δ δ + δ 4 b k δ i.e..0 δ δ + δ 4 a k δ a k δ a 0 δ a 1 δ. δ + δ 4 a N δ + b N δ 1 δ 1 a N δ + δ b N δ δ 3 δ + δ + δ 3 + δ 4 δ δ δ a N δ 1 + δ b N δ b k δ δ a N δ 1 + δ b N δ.

14 Roman Witu la Next form we obtain S 1N δ a k δ + Thus by.19.0 we derive b k δ 1 δ δ k. δ δ + δ 4 S 1N δ δ 1 a N δ 1 + δ b N δ + δ a N δ 1 + δ b N δ δ δ δ + δ N 1 4 δ 6 a N δ δ b N δ + δ + δ 4 δ + 1 which is exactly the formula.8. References 1 δ δ N 1 [1] Knott R. Fibonacci Golden Ratio Formulae. ac.uk/hosted-sites/r.knott/fibonacci/fibformulae.html [] Koshy T. Fibonacci Lucas Numbers with Application. Wiley New York: 001. [3] Witu la R. S lota D. Warzyński A. Quasi-Fibonacci Numbers of Seventh Order. J. Integer Seq Article [4] Witu la R. S lota D. Quasi-Fibonacci Numbers of Order 11. J. Integer Sequences Article [] Witu la R. Jama D. Connections Between the Primitive -th Roots of Unity Fibonacci Numbers Part I. Zeszyty Nauk. Pol. Sl. Mat.-Fiz [6] Witu la R. S lota D. δ-fibonacci Numbers. Appl. Anal. Discrete Math [7] Witu la R. Jama D. Connections Between the Primitive -th Roots of Unity Fibonacci Numbers Part II. Zeszyty Nauk. Pol. Sl. Mat.-Fiz [8] Witu la R. S lota D. Quasi-Fibonacci Numbers of Order 13 on the Occasion the Thirteenth International Conference on Fibonacci Numbers Their Applications. Congr. Numer Received by the editors December 9 010

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