On the complex k-fibonacci numbers

Transcription

1 Falcon, Cogent Mathematics 06, 3: APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE On the complex k-fibonacci numbers Sergio Falcon * ceived: 9 January 05 Accepted: 0 May 06 First Published: 0 June 06 *Corresponding author: Sergio Falcon, Department of Mathematics, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain sergiofalcon@ulpgces viewing editor: Benchawan Wiwatanapataphee, Curtin University, Australia Additional information is available at the end of the article Abstract: We first study the relationship between the k-fibonacci numbers and the elements of a subset of Q Later, and since generally studies that are made on the Fibonacci sequences consider that these numbers are integers, in this article, we study the possibility that the index of the k-fibonacci number is fractional; concretely, n+ In this way, the k-fibonacci numbers that we obtain are complex And in our desire to find integer sequences, we consider the sequences obtained from the moduli of these numbers In this process, we obtain several integer sequences, some of which are indexed in The Online Enciplopedy of Integer Sequences OEIS Subects: Applied Mathematics; Mathematics & Statistics; Science; Theory of Numbers Keywords: k-fibonacci numbers; Binet identity; complex numbers AMS subect classifications: 5A36; C0; B39 Introduction Classical Fibonacci numbers have been very used in different sciences such biology, demography, or economy Hoggat, 969; Koshy, 00 cently, they have been applied even in high-energy physics El Naschie, 00, 006 But, there exist generalizations of these numbers given by such researchers ABOUT THE AUTHOR Sergio Falcon Santana obtained his PhD in Mathematics from ULPGC He is a professor in the University of Las Palmas de Gran Canaria in Mathematics, Advanced Calculus, Numerical Calculus, Algebra He is an author of several books on mathematics for technical schools Despite getting only about years, dedicated to the University on time complete, at this time, he has published more than 70 articles in mathematics covering various aspects such as disclosure mathematics mathematical didactics and, above all, mathematical research of top level Some of his research articles have been published in the best mathematics ournals around the world He has some numerical successions indexed in The On-Line Encyclopedia of Integer Sequences of NJA Sloane and have helped to improve some aspects of others He is a partner of the al Sociedad Matemática Española RSME and the Association of teachers of Mathematics Puig Adam He is a member of the University Institute of Applied Microelectronics PUBLIC INTEREST STATEMENT So far, the Fibonacci numbers have been generalized in different forms one being that made by ourselves and we call k-fibonacci numbers But all these generalizations have been in the real numbers field What I propose in this study is to extend the area of Fibonacci numbers to the complex numbers field 06 The Authors This open access article is distributed under a Creative Commons Attribution CC-BY 40 license Page of 9

2 Falcon, Cogent Mathematics 06, 3: as Horadam 96 and recently by Bolat and Kse 00, Ramírez 05, Salas 0 and the current author Falcon and Plaza 007a, 007b, 009a In this paper, this last generalization is presented, so called the k-fibonacci numbers On the k-fibonacci numbers { For any positive real number k, the k-fibonacci sequence, say n, is defined recurrently by }n N n+ kn + n for n with initial conditions 0 0 and For k, the classical Fibonacci sequence is obtained and for k, the Pell sequence appears The well-known Binet formula Falcon & Plaza, 007a; Horadam, 96; Spinadel, 00 allows us to relate the k-fibonacci numbers to the characteristic roots σ and σ associated to the recurrence relation r kr+ so that n σn σn σ σ If σ denotes the positive characteristic root, σ k+ k +4, the general term may be written as n σn σ n, and it is verified that the limit of the quotient of two terms of the sequence σ + σ F {n } n N is lim k,n n σ r n In particular, if k, then σ is the Golden Ratio, φ + 5 ; if k, σ is the Silver Ratio and for k 3, we obtain the Bronze Ratio Spinadel, 00 Among other properties that we can see in Falcon and Plaza 007a, 007b, 009a, we will need the Simson Identity: n n+ F k,n n The k-fibonacci numbers and the set {a, b, a, b Q} Let us consider the set {a, b, a, b Q} C In, we define the operations a, b+c, d a + c, b + d and for a fixed number k N {0}, a, b c, d ad+ bc+ kac, ac+ bd Then, is an abelian field, with the identity element being 0,, and a, b a, ka+ b k ab + b a the inverse of the element a, b 0, 0 From the definition of sum, it follows that na, b na, nb, for n Q and a, b a, b a, b The k-fibonacci numbers and the pairs, 0 n Now, we consider the subset A, defined as {, 0 n, n,, } The elements of are related to the k-fibonacci numbers in the following form Lemma The elements of are of the form, 0 n n, n 3 Proof We proceed by induction on n For n, it is, 0, 0 Assume that, 0 n n, n holds Then,, 0 n+ n, n, 0 n + kn, n n+, n Page of 9

3 Falcon, Cogent Mathematics 06, 3: It follows that, from Equations and, we can deduce that, 0 n n n, n This formula allows us to define the k-fibonacci numbers of negative index as is known, n n n 4 On the other hand,, 0 n, 0 n, 0 0 and taking into account the Simson Identity,, 0 n, 0 n n, n n, n n n + n n + kn n, n n + n n n n n+ n n kf, k,n F + F F k,n k,n k,n+ n n n+ n n+, n 0, Consequently, we can define, 0 0 0,, with 0, being the multiplicative identity in From Formula, it is,, k +, and consequently, k n, n n,, k + n k The previous definition and the results obtained allow us to find some properties of the k-fibonacci numbers, previously proven in papers Falcon & Plaza, 007a, 007b, 009a, in the next subsection Convolution of the k-fibonacci numbers It is obvious that, 0 m, 0 n, 0 m+n m+n, m+n 5 On the other hand, and taking into account the Simson Identity,, 0 m, 0 n m, m n, n m n + m n + km n, m n + m n m kn + n +m n, m n + m n m n+ + m n, m n + m n Equating first elements of this pair with 5, we deduce the convolution formula Falcon & Plaza, 007b; Vada, 989: m+n m n+ + m n And as particular cases, we will mention the following: If m n, then n k n+ n If m n +, then n+ F k,n+ + n 3 k-fibonacci numbers of the half index In this section, we will study the k-fibonacci numbers of the half index We will call n+ a k-fibonacci number of the half index Taking into account, 0, 0, 0 and Equation, it is,,, 0 Hence, applying definition:, 0 + kf, k, F + k, F k, k + +, F + k, F k, 3 +, F + k, F k, Page 3 of 9

4 Falcon, Cogent Mathematics 06, 3: Then, we obtain the system of quadratic equations 3 + F k, From the second equation, we obtain ±i If we suppose the real part of the complex number is positive, then we must take i placing in 6, the following equation holds: 3 i 7 Hence, we can accept for the k-fibonacci numbers of index n+ the following definition: n+ n+ if k, n+ This formula is very similar to Formula 4 for the k-fibonacci numbers of negative integer indices 3 Binnet identity The Binnet identity for the k-fibonacci numbers of integer indices Falcon & Plaza, 007b continues being valid for the case of that n r+ because its characteristic equation is the same in both cases, r kr 0 This shows that we could have defined the k-fibonacci numbers from this formula and then to then found the different sequences for k,, It is noteworthy that many of the general formulas found for the k-fibonacci numbers continue checking for the case of n r +, except perhaps that sometimes it is necessary to multiply by the factor i We have proved in Falcon and Plaza 007a, 007b and in this same paper, the formulas n+ F + k,n+ n and F k,n k F k,n+ n Next, we will prove that both formulas are also valid for any number if we take into account the number of the half index From the preceding formulas,, 0 n+, 0 n+, 0 n+ n+ n+, n n n+, n From the second terms of both pairs, n F k, n+ And from the first terms, n+, n + kf, F + F k, n+ k, n+ k, n n+, n + F k, n n+ n+ n+ k n+3 n k n+ n+ k + kf k, n+ n + n F k, n+3 n+3 F k, n + n + n Finally, if we substitue n by n in these formulae, we find both initial formulas Also the convolution formula remains valid, and its proof is similar to the preceding, from, 0 n+m, 0 n, 0 m+ and we would obtain: n+m n+ F k, m+ + n m Page 4 of 9

5 Falcon, Cogent Mathematics 06, 3: However, the Catalan formula for the k-fibonacci numbers dictates that if n is an integer number Falcon & Plaza, 007a, 007b, 009a, then n r n F k,n n r F k,r changes if the number is of the half index In this case, the Catalan formula takes the form n+ n r if k,r It is enough to apply the Binnet Identity, taking into ac- F r k, n+ F k, n+ count that σ σ Consequently, the Simson Identity n n+ F k,n n is transformed into n n+3 F k, n+ n i 3 Some notes about the k-fibonacci numbers of the half index Both al and imaginary parts of n+ Gaussian Integer Weisstein, 009 never are integers Consequently, n+ never is a For a fixed number k, it is verified that n < F k, n+ < n+ except for the classical Fibonacci number F 3 < F 3 Taking into account that σ decreases when n increases, the absolute value of n+ tends to the al part of this number: lim n n+ lim n n+ 33 Another formula for the k-fibonacci numbers of the half index n+ σ σ n+ If, in the Binnet Identity n+, we multiply both numerator and denominator of the σ σ fraction by σ n+ + σ n+ and then we do the division, on obtaining the following formula for the calculus of the k-fibonacci number of the half index: 8 n+ 4 On the sequences of k-fibonacci numbers of half index{ Let us consider the k-fibonacci sequence of complex numbers The Binnet Identity can be indicated as n+ hence n+ σ n+ Consequently, n+ + σ n+ n σ n σ n σ + n k + 4 σ n σ i σ n σ k + 4 n+ } n+ σ n+ σ n+ k + 4 n N 9 0 Im n+ n σ σk n + 4 σ Hence, the real part of the first term of this sequence is and the real parts of k + 4 the successive terms is obtained multiplying the real part of the previous term by σ Similarly, the imaginary part of the first term of this sequence is Im and the imaginary parts of k +4σ the successive terms are obtained by multiplying the imaginary part of the previous term by σ σ Page 5 of 9

6 Falcon, Cogent Mathematics 06, 3: Consequently, this k-fibonacci sequence takes the form { } σ n+ + n σ i n k + 4σ { From Equations 8 and 0, we deduce that the sequence From Equation 9, we obtain the following interesting results n+ } diverges 4 Theorem For all n N, following equalities hold: Im Im n+ n+ n+ n+ σ r n+ 3 lim n r σ r n+ σ r The first two formulae are obvious As for the third, we must bear in mind that the imaginary part tends to zero when the index tends to infinite, so its contribution to the modulus of the complex number decreases when n increases In consequence, lim n n+ n+ n lim n+ n+ n+ n+ The next theorem relates the k-fibonacci numbers of the half index to the k-fibonacci numbers of the integer index σ r 4 Theorem For all integers k and for all n N: n+ n+3 if k, n+ n+ n+ in +i n r Proof Applying the Binnet identity to both sides of this equation, taking into account σ σ i, and after removing k + 4 from both denominators, it becomes LHS σ n+ σ n+ σ n+ + σ n+ + iσ n+ + iσ n+ σ n+ + n iσ r+ iσ n+ + n σ r + i n σ r+ + σ n+ + n σ r iσn+ RHS σ n+ σ n+ σ n+ σ n+ iσ n + iσ n + n iσ r σr σ σ σ n+ n+ σ r iσn+ + i n+ σ r n+ σ r + σn+ + i n+ σ r iσ n+ + n iσ r+ + n iσ r + n iσ r + n iσ r+ LHS In particular, for n r 0, Equation 7 is obtained Page 6 of 9

7 Falcon, Cogent Mathematics 06, 3: On the sequences of k-fibonacci numbers of the half index Taking into acccount a k-fibonacci number of the half index is a complex number which both realpart and imaginary part are never integers, the sequences of these numbers do not have greater interest Of course, in these sequences, the initial relation is verified, that is n+ kn + n The sequences related to the modulus of these complex numbers are more interesting Let r be the modulus of the k-fibonacci number r, when r n+ The floor function of n is the integral part of this number: O k,n Floor[ n ] We can also say that they are obtained by the rounding down of n The round function of n is the closest integer to this number: R k,n Round[ n ] The ceiling function of n is the function whose value is the smallest integer, not less than n : C k,n Ceiling[ n ] We can also say that they are obtained by the rounding up of n For k,, 3, we will obtain the following sequences, none of which is indexed in Sloane 006, from now on OEIS: For k : a O {0, 0,,, 3, 6, 0, 6, } b R {,,, 4, 6, 0, 7, 7, } c C {,,, 3, 4, 7,, 7, 7, } For k : a O {0,, 3, 7, 8, 45, 08, 6, } b R {,, 3, 8, 9, 45, 09, 63, } c C {,, 4, 8, 9, 46, 09, 63, } 3 For k 3: a O 3 {0,, 5, 8, 59, 98, 654, } b R 3 {,, 5, 8, 60, 98, 654, } c C 3 {,, 6, 9, 60, 99, 655, } 5 Integer sequences from, n Let us remember that 0, is the unity element of {a, b}, so 0, n 0, Then, taking into account Equation 3,, 0 n n, n, if a and b are non-null simultaneously, then n a, b n a, 0+b0, n n a, 0 n b n 5 Expression of a k-fibonacci number whose index is a multiple of another index As, 0 n m, 0 n m n m, n m and m, 0 n m n, n m m F m k,n F F, n k,m k,m, we obtain n a n b, 0 n n n a n b F k,n, n Page 7 of 9

8 Falcon, Cogent Mathematics 06, 3: m n m For m : n For m 3: 3n Plaza, 007a, 007b m F m k,n F n k,m F k,n F n k, kf + F n k,n k,n Falcon & Plaza, 007a, 007b 3 F 3 k,n F n k,3 k + 4 F 3 3kn k,n F + k,n 3k F n k,n Falcon & m Equation 3 can be written as m n n m n is a multiple of n In short: n If a 0 and b 0: a, b n n a n b F k,n, n If b 0: a,0 n a n, 0 n a n n, n m m F F n k,n k,m and that denotes that 3 If a 0: 0, b n b n 0, n b n 0, n If a b, then, n n F k,n, n : the first terms of the sequence of powers {, n } are the binomial transforms of the k-fibonacci sequence Falcon & Plaza, 009b 5 Integer sequences of coefficients from, n In the sequel, we give the expressions of the first terms of this sequence {, n }, for n 0,,,, 0 0,,,,,, k +,, 3 k +,, k + 3k + 4, k + 4, 4 k + 3k + 4, k + 4, k 3 + 4k + 8k + 8, k + 4k + 8 With the coefficients of the first terms of the pairs of the Second-Hand Side, we form Table 3 The number a i, is the coefficient of k i in the first elements of the pairs of, i The first diagonal is the sequence of powers of, { n }, n 0,,, n n Table Coefficients in Fk,n Page 8 of 9

9 Falcon, Cogent Mathematics 06, 3: Any other coefficient of the row r and column c, can be calculated as a r,c a r,c + instance, c a r,c For Moreover: the sequence of the sums of the coefficients of each row is the bisection of the classical Fibonacci sequence {, 3, 8,, 55, 44, 377, 987, } A00906 and its alternate sums is this same sequence {,,, 3, 5, 8, 3,, } A Hence, we can write n n Fn F n and n n n Fn F n Only the first four column sequences are listed in OEIS as A0000, A00007, A034856, A00646 Each diagonal sequence is the convolution of the preceding diagonal sequence and A078 {,,, 4, 8, 6, 3, } and are listed in OEIS: A000079, A0079, A0496, A0496, A055589, A05585, A055853, A055854, and A Finally, we indicate that the generating function of the diagonal sequence D n {, n, } is xn 4 dn x n Funding The author received no direct funding for this research Author details Sergio Falcon sergiofalcon@ulpgces ORCID ID: Department of Mathematics, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain Citation information Cite this article as: On the complex k-fibonacci numbers, Sergio Falcon, Cogent Mathematics 06, 3: 0944 ferences Bolat, C, & Kse, H 00 On the properties of k-fibonacci numbers International Journal of Contemporary Mathematical Sciences, 5, El Naschie, M S 00 Notes on superstrings and the infinite sums of Fibonacci and Lucas numbers Chaos, Solitons & Fractals,, El Naschie, M S 006 Topics in the mathematical physics of E-infinity theory Chaos, Solitons & Fractals, 3, Falcon, S, & Plaza, Á 007a On the Fibonacci k-numbers Chaos, Solitons & Fractals, 3, Falcon, S, & Plaza, Á 007b The k-fibonacci sequence and the Pascal -triangle Chaos, Solitons & Fractals, 33, Falcon, S, & Plaza, Á 009a k-fibonacci sequences modulo m Chaos, Solitons & Fractals, 4, Falcon, S, & Plaza, Á 009b Binomial transforms of the k-fibonacci sequence International Journal of Nonlinear Sciences & Numerical Simulation, 0, Hoggat, V E 969 Fibonacci and Lucas numbers Boston, MA: Houghton-Mifflin Horadam, A F 96 A generalized Fibonacci sequence Mathematics Magazine, 68, Koshy, T 00 Fibonacci and Lucas numbers with applications New York, NY: A Wiley-Interscience Ramírez, J L 05 Some combinatorial properties of the k- Fibonacci and k-lucas Quaternions Analele stiintifice ale Universitatii Ovidius Constanta, 3, 0 Salas, A 0 About k-fibonaci numbers and their associated numbers International Mathematical Forum, 50, Sloane, N J A 006 The on-line encyclopedia of integer sequences trieved from wwwresearchattcom/nas/ sequences/ Spinadel, V W 00 The metallic means family and forbidden symmetries International Journal of Mathematics,, Vada, S 989 Fibonacci & Lucas numbers, and the golden section Theory and applications Mineola, NY: Ellis Horwood Weisstein, E W 009 Gaussian integer, From MathWorld-A Wolfram Web resource trieved from wolframcom/gaussianintegerhtml 06 The Authors This open access article is distributed under a Creative Commons Attribution CC-BY 40 license You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially The licensor cannot revoke these freedoms as long as you follow the license terms Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits Page 9 of 9