DOCSLIB.ORG

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 34 No

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
International Journal of Pure and Applied Mathematics ————————————————————————– Volume 34 No International Journal of Pure and Applied Mathematics ————————————————————————– Volume 34 No. 2 2007, 267-278 GENERATING MATRICES FOR FIBONACCI, LUCAS AND SPECIAL ORTHOGONAL POLYNOMIALS WITH ALGORITHMS 1 2 3 Mustafa A¸sci §, Bayram C¸ekım , Dursun Ta¸sci 1,2,3Department of Mathematics Faculty of Sciences and Arts University of Gazi Teknikokullar, Ankara, 06500, TURKEY 1e-mail: [email protected] 2e-mail: [email protected] 3e-mail: [email protected] Abstract: In this paper we get the Fibonacci polynomials for special cases of Sturm Liouville boundary value problems. We show the orthogonallity of Fibonacci polynomials. In Section 3 by using the definition of n n Hessenberg × matrices we give the generating matrices of the Fiboniacci and Lucas polyno- mials and for the special orthogonal polynomials. In the last section we give two algorithms for finding the Fibonacci and Lucas polynomials. AMS Subject Classification: 15A15, 11B39, 15A36 Key Words: Fibonacci polynomial, Sturm Liouville algorithm 1. Introduction The Fibonacci sequence, Fn , is defined by the recurrence relation, for n 1 { } ≥ Fn+1 = Fn + Fn 1 , (1.1) − where F = F = 1. The Lucas sequence, Ln , is defined by the recurrence 0 1 { } Received: November 2, 2006 c 2007, Academic Publications Ltd. §Correspondence author 268 M. A¸sci, B. C¸ekım, D. Ta¸sci relation, for n 1 ≥ Ln+1 = Ln + Ln 1 , (1.2) − where L0 = 2, L1 = 1. Large classes of polynomials can be defined by Fibonacci-like recurrence relation, and yield Fibonacci numbers. Such polynomials, called the Fibonacci polynomials, were studied in 1883 by the Belgian mathematician Eugene Charles Catalan and the German mathematician E. Jacobsthal. The polynomials fn(x) studied by Catalan are defined by the recurrence relation fn(x)= xfn 1(x)+ fn 2(x) , (1.3) − − where f (x) = 1, f (x)= x, and n 3. The Fibonacci polynomials studied by 1 2 ≥ Jacobsthal were defined by Jn(x)= Jn 1(x)+ xJn 2(x) , (1.4) − − where J1(x)=1= J2(x). Lucas polynomials Ln(x), originally studied in 1970 by Bicknell, are defined by Ln(x)= xLn 1(x)+ Ln 2(x) , (1.5) − − where L (x) = 2, L (x)= x and n 2. 0 1 ≥ A differential equation defined on the interval a x b having the form ≤ ≤ of d dy p(x) + [q(x)+ λr(x)] y = 0 dx dx and the boundary conditions a1y(a)+ a2y′(a) = 0 , b1y(b)+ b2y′(b) = 0 , is called as Sturm-Liouville boundary value problem or Sturm-Liouville system where p(x) > 0, q(x), the weighting function r(x) > 0 are given functions; a1, a2, b1, b2 are given constants; and the eigenvalue λ is an unspecified param- eter. For a = 1, b = 1, p(x) = √1 x2, q(x) = 0, r(x) = 1 and λ = n2 − − √1 x2 the Sturm-Liouville equation becomes the Chebyshev’s differential− equation 2 2 (1 x )y′′ xy′ + n y = 0 , − − which is defined on 1 <x< 1. The solutions of the Chebyshev’s differential − equation with n = 0, 1, 2, 3, ... is called Chebyshev polynomials Tn(x) which GENERATING MATRICES FOR FIBONACCI... 269 form a complete orthogonal set on the interval 1 <x< 1 with respect to − r(x)= 1 √1 x2 − −x2 −x2 For a = , b = , p(x)= e 2 , q(x) = 0, r(x)= e 2 and λ = 2n the −∞ ∞ Sturm-Liouville equation becomes the Hermite’s differential equation y′′ 2xy′ + 2ny = 0 , − which is defined on <x< . The solutions of the Hermite’s differential −∞ ∞ equation with n = 0, 1, 2, 3, ... is called Hermite polynomials Hn(x) which form a complete orthogonal set on the interval <x< with respect to r(x)= −x2 −∞ ∞ e 2 We can get Legendre polynomials, Bessel functions, Laguerre polynomials for some special cases of the Sturm-Liouville boundary value problem. 2. Computations on Fibonacci Differential Equation The ordinary differential equation 2 (x + 4)y′′ + 3xy′ + λy = 0 (2.1) is known as Fibonacci differential equation. When we solve the equation the solutions are the Fibonacci polynomials. Let us show that the eigenvalue λ is (1 n2): − ∞ n By the serial solution of ODE, suppose that y = anx then y′ = n=0 ∞ n 1 ∞ n 2 nanx − and y′′ = n(n 1)anx − . n=0 n=0 − Substituting these values in (2.1) 2 ∞ n 2 ∞ n 1 ∞ n (x + 4) n(n 1)anx − + 3x nanx − + λ anx = 0 , − n=0 n=0 n=0 ∞ n ∞ n 2 ∞ n 1 n(n 1)anx + 4 n(n 1)anx − + 3x nanx − − − n=0 n=2 n=0 ∞ n + λ anx = 0 , n=0 ∞ n ∞ n ∞ n n(n 1)anx + 4 (n + 1)(n + 2)an x + 3 nanx − +2 n=0 n=0 n=0 270 M. A¸sci, B. C¸ekım, D. Ta¸sci ∞ n + λ anx = 0 , n=0 ∞ n (n(n 1)an + 4(n + 1)(n + 2)an + 3nan + λan)x = 0 . − +2 n=0 For this solution we get (n2 + 2n + λ) an = − an , n 0 , +2 4(n + 1)(n + 2) ≥ or (n2 1+ λ) an+1 = − − an 1 , n 1 , 4n(n + 1) − ≥ then we get λ = 1 n2 − When we get λ = 1 n2 the solutions of the ODE (x2 + 4)y + 3xy + (1 − ′′ ′ − n2)y = 0 are the Fibonacci polynomials. Now we show that Fibonacci polynomials are orthogonal with respect to the weight function √2 x2 + 4. Theorem 1. Let Fn and Fm be Fibonacci polynomials as defined in (1.3). Then we get 2i 2 2 0, m = n , x + 4FmFndx = n+1 2i ( 1) 2iπ, m = n . − − Proof. Case 1. m = n. Since Fm and Fn are the solutions of ODE (2.1) 2 2 (x + 4)F ′′ + 3xF ′ + (1 n )Fn = 0 . n n − 2 2 In this equation product this equation with √x + 4Fm 2 2 (x + 4)F ′′ + 3xF ′ + (1 n )Fm = 0 m m − 2 2 multiply this equation with √x + 4Fn 3 2 2 2 2 2 2 2 (x + 4) F ′′Fm + 3x x + 4FmF ′ + (1 n ) x + 4FmFn = 0 , (2.2) n n − 3 2 2 2 2 2 2 2 (x + 4) Fm′′ Fn + 3x x + 4FnFm′ + (1 m ) x + 4FmFn = 0 . (2.3) − Subtract (2.2) and (2.3) we get 3 2 2 2 2 (x + 4) F ′′Fm F ′′ Fn + 3x x + 4 FmF ′ FnF ′ n − m n − m GENERATING MATRICES FOR FIBONACCI... 271 2 2 2 2 x + 4FmFn(n m ) = 0 , − − d 3 2 2 2 2 2 2 (x + 4) FmF ′ FnF ′ = x + 4FmFn(n m ) . dx n − m − Integrate this equations from 2i to 2i − 2i 2i d 3 2 2 2 2 2 2 (x + 4) FmFn′ FnFm′ dx = (n m ) x + 4FmFndx , 2i dx − − 2i − − 2i 3 2i 2 2 2 2 2 2 (x + 4) FmFn′ FnFm′ = (n m ) x + 4FmFndx , − 2i − 2i − − 2i 2 2 2 2 (n m ) x + 4FmFndx = 0 , − 2i − when n = m we get 2i 2 2 x + 4FmFndx = 0 . 2i − Case 2. m = n. This case is computed by a short MATLAB algorithm easily. 3. Generating Matrices for Fibonacci, Lucas Polynomials and Special Functions A lower Hessenberg matrix, A, is an n n matrix, where aj,k = 0 whenever × k > j +1 and aj,j = 0 for some j. That is, all entries above the superdiagonal +1 are 0 but the matrix is not lower triangular. Throughout this paper we will refer to the following lower Hessenberg matrix m1,1 m1,2 0 ... ... 0 .. .. m2,1 m2,2 m2,3 . 0 .. .. m , m , m , . 0 M = 3 1 3 2 3 3 . (3.1) n . . .. .. .. 0 . .. .. .. . mn 1,n − mn,1 mn,2 ... ... mn,n 1 mn,n − We will consider the sequence det Mn,n 1 . Our result is stated in the { ≥ } following theorem. 272 M. A¸sci, B. C¸ekım, D. Ta¸sci Theorem 2. Let Mn be as above for all n 1 and define det M = 1 ≥ 0 Then det Mn,n 0 satisfies: { ≥ } det M0 = 1, det M1 = m1,1 and n 1 n 1 − n r − det Mn = mn,n. det Mn 1 + ( 1) − mn,r mj,j+1. det Mr 1 , − − − r=1 j=r n 2. ≥ Proof. It is proved in [2]. We can obtain the family of matrices Fn by letting mj,j = 1 and mj,j = { } +1 mj ,j = i = √ 1, 1 j n 1 in the above theorem. Then det M = 1, +1 − ≤ ≤ − 1 det M2 = 2 and det Mn = det Mn 1 + det Mn 2, which is exactly the Fibonacci − − recurrence. By using above theorem we get the following results by determinant com- putations. Corollary 1. Let Fn be the n n Hessenberg matrix defined as in (3.1). × For special cases as below we get det Fn = Fn(x) 1 0 0 ... ... 0 . 0 x i 0 .. 0 . 0 i x i .. 0 Fn = , .. .. 0 0 i . 0 . .. .. .. . i 0 0 ... 0 i x where Fn(x) is the n-th Fibonacci polynomial. Corollary 2. Let Ln be the n n Hessenberg matrix defined as in (3.1). × For special cases as below we get det Ln = Ln(x) 2 0 0 ... ... 0 . x .. 0 2 i 0 0 . 0 i x i .. 0 Ln = , .. .. 0 0 i . 0 . .. .. .. . i 0 0 ... 0 i x GENERATING MATRICES FOR FIBONACCI... 273 where Ln(x) is the n-th Lucas polynomial. Corollary 3. Let Hn be the n n Hessenberg matrix defined as in (3.1). × For m , = 1, mi,i = 2x, i = 1, mi,i = √2i, and mi ,i = (n 2)√2i.
Load more

Recommended publications
  • Fibonacci and -Lucas Numbers
    Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 107145, 7 pages http://dx.doi.org/10.1155/2013/107145 Research Article Incomplete -Fibonacci and -Lucas Numbers José L. Ramírez Instituto de Matematicas´ y sus Aplicaciones, Universidad Sergio Arboleda, Colombia Correspondence should be addressed to JoseL.Ram´ ´ırez; [email protected] Received 10 July 2013; Accepted 11 September 2013 Academic Editors: B. Li and W. van der Hoek Copyright © 2013 JoseL.Ram´ ´ırez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers. 1. Introduction In [6],theauthorsdefinetheincompleteFibonacciandLucas -numbers. Also the authors define the incomplete bivariate Fibonacci numbers and their generalizations have many Fibonacci and Lucas -polynomials in [7]. interesting properties and applications to almost every field On the other hand, many kinds of generalizations of of science and art (e.g., see [1]). Fibonacci numbers are Fibonaccinumbershavebeenpresentedintheliterature.In defined by the recurrence relation particular, a generalization is the -Fibonacci Numbers. For any positive real number ,the-Fibonacci sequence, 0 =0, 1 =1, +1 = +−1,⩾1. (1) say {,}∈N, is defined recurrently by There exist a lot of properties about Fibonacci numbers. =0, =1, = + ,⩾1. In particular, there is a beautiful combinatorial identity to ,0 ,1 ,+1 , ,−1 (4) Fibonacci numbers [1] In [8], -Fibonacci numbers were found by studying ⌊(−1)/2⌋ −−1 = ∑ ( ). the recursive application of two geometrical transformations (2) =0 used in the four-triangle longest-edge (4TLE) partition.
    [Show full text]
  • On H(X)-Fibonacci Octonion Polynomials, Ahmet Ipek & Kamil
    Alabama Journal of Mathematics ISSN 2373-0404 39 (2015) On h(x)-Fibonacci octonion polynomials Ahmet Ipek˙ Kamil Arı Karamanoglu˘ Mehmetbey University, Karamanoglu˘ Mehmetbey University, Science Faculty of Kamil Özdag,˘ Science Faculty of Kamil Özdag,˘ Department of Mathematics, Karaman, Turkey Department of Mathematics, Karaman, Turkey In this paper, we introduce h(x)-Fibonacci octonion polynomials that generalize both Cata- lan’s Fibonacci octonion polynomials and Byrd’s Fibonacci octonion polynomials and also k- Fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula and and generating function of h(x)-Fibonacci octonion polynomial sequence. Introduction and Lucas numbers (Tasci and Kilic (2004)). Kiliç et al. (2006) gave the Binet formula for the generalized Pell se- To investigate the normed division algebras is greatly a quence. important topic today. It is well known that the octonions The investigation of special number sequences over H and O are the nonassociative, noncommutative, normed division O which are not analogs of ones over R and C has attracted algebra over the real numbers R. Due to the nonassociative some recent attention (see, e.g.,Akyigit,˘ Kösal, and Tosun and noncommutativity, one cannot directly extend various re- (2013), Halici (2012), Halici (2013), Iyer (1969), A. F. Ho- sults on real, complex and quaternion numbers to octonions. radam (1963) and Keçilioglu˘ and Akkus (2015)). While ma- The book by Conway and Smith (n.d.) gives a great deal of jority of papers in the area are devoted to some Fibonacci- useful background on octonions, much of it based on the pa- type special number sequences over R and C, only few of per of Coxeter et al.
    [Show full text]
  • Alyssa Farrell
    Recursive Patterns and Polynomials Alyssa Farrell Iowa State University MSM Creative Component Summer 2006 Heather Thompson, Major Professor Irvin Hentzel, Major Professor Patricia Leigh, Committee Member Farrell 2 Patterns have been studied for centuries. Some of these patterns have been graced with names of the people who are most remembered for their work with them or related topics. Among these are the Fibonacci numbers, Lucas numbers, Pascal’s Triangle, Chebyshev’s Polynomials, and Cardan Polynomials. Patterns are used as building blocks in education from newborn babies and young children all the way through high school and beyond into post-secondary work. We were introduced as early as infancy to learning to recognize patterns, meaning both similarities and differences, when learning colors, shapes, the alphabet, and numbers. As we grew, we were introduced to new mathematical patterns like learning to count by 2’s, or 5’s. As we were more formally introduced to shapes, we learned to recognize that a square has four sides the same length, but unlike a kite, the square has four angles that are 90 degrees. Then we learned to add, subtract, multiply, and divide. All of these necessary operations were taught using a pattern-approach. Do you remember learning to read? We were taught to recognize patterns of letters which made certain sounds and that when certain sounds were strung together they made words. These patterns continued to be used as we developed higher in our math education. Teachers in junior high or middle school introduced us to an abstract idea of letting letters represent numbers as variables.
    [Show full text]
  • Certain Combinatorial Results on Two Variable Hybrid Fibonacci Polynomials
    International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 603–613 © Research India Publications http://www.ripublication.com/ijcam.htm Certain Combinatorial Results on Two variable Hybrid Fibonacci Polynomials R. Rangarajan, Honnegowda C.K.1, Shashikala P. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru - 570 006, INDIA. Abstract Recently many researchers in combinatorics have contributed to the development of one variable polynomials related to hypergeometric family. Here a class of two variable hybrid Fibonacci polynomials is defined and proved certain combinatorial results. AMS subject classification: 05A19, 11B39 and 33C45. Keywords: Combinatorial Identities, Fibonacci Numbers and Hyper Geometric Functions in one or more Variables. 1. Introduction The natural and the most beautiful three term recurrence relation is Fn+1 = Fn + Fn−1, F1 = 1,F2 = 1, where Fn is the nth Fibonacci number [1, 2, 3, 13]. There are many one variable generalizations of Fn related to hypergeometric family [3, 4]. Two important such generalizations are Catalan polynomials and Jacobsthal polynomials denoted by (C) (J ) fn (x) and fn (x) which exhibits many interesting combinatorial properties [4, 6, 7, 8, 9, 10, 12]. 1Coressoponding author. 604 R. Rangarajan, Honnegowda C.K., and Shashikala P. Two hypergeometric polynomials, namely, Un(x) and Vn(x) called Tchebyshev poly- nomials of second and third kind are given by n+1 n+1 1 2 2 Un(x) = √ x + x − 1 − x − x − 1 2 x2 − 1 3 1 − x = (n + 1) F −n, n + 2; ; 2 1 2 2 and = − Vn(x) Un+1(x) Un(x) 1 1 − x = F −n, n + 1; ; 2 1 2 2 which have many applications in both pure and applied mathematics [5, 6, 7, 8, 9, 10, 3 3 11, 12].
    [Show full text]
  • Generalized Fibonacci-Type Polynomials
    GENERALIZED FIBONACCI-TYPE POLYNOMIALS Yashwant K. Panwar 1 , G. P. S. Rathore 2 , Richa Chawla 3 1 Department of Mathematics, Mandsaur University, Mandsaur, (India) 2 Department of Mathematical Sciences, College of Horticulture, Mandsaur,(India) 3 School of Studies in Mathematics, Vikram University, Ujjain, (India) ABSTRACT In this study, we present Fibonacci-type polynomials. We have used their Binet’s formula to derive the identities. The proofs of the main theorems are based on simple algebra and give several interesting properties involving them. Mathematics Subject Classification: 11B39, 11B37 Keywords: Generalized Fibonacci polynomials, Binet’s formula, Generating function. I. INTRODUCTION Fibonacci polynomials are a great importance in mathematics. Large classes of polynomials can be defined by Fibonacci-like recurrence relation and yield Fibonacci numbers [15]. Such polynomials, called the Fibonacci polynomials, were studied in 1883 by the Belgian Mathematician Eugene Charles Catalan and the German Mathematician E. Jacobsthal. The polynomials fxn ()studied by Catalan are defined by the recurrence relation fn21()()() x xf n x f n x (1.1) where f12( x ) 1, f ( x ) x , and n 3 . Notice that fFnn(1) , the nth Fibonacci number. The Fibonacci polynomials studied by Jacobsthal were defined by Jn()()() x J n12 x xJ n x (1.2) where J12( x ) 1 J ( x ) , and . The Pell polynomials pxn ()are defined by pn( x ) 2 xp n12 ( x ) p n ( x ) (1.3) where p01( x ) 0, p ( x ) 1, and n 2 . The Lucas polynomials lxn (), originally studied in 1970 by Bicknell, are defined by ln()()() x xl n12 x l n x (1.4) where l01( x ) 2, l ( x ) x , and n 2 .
    [Show full text]
  • Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 March 2017 doi:10.20944/preprints201703.0208.v1 Peer-reviewed version available at Bulletin of the Iranian Mathematical Society 2019, 45, 1821-1829; doi:10.1007/s41980-019-00232-4 CLOSED EXPRESSIONS OF THE FIBONACCI POLYNOMIALS IN TERMS OF TRIDIAGONAL DETERMINANTS FENG QI Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China JING-LIN WANG Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China BAI-NI GUO School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Abstract. In the paper, the authors find a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant. 1. Main results A tridiagonal matrix (determinant) is a square matrix (determinant) with nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal. See the paper [14] and closely-related references therein. A square matrix H = (hij)n×n is called a tridiagonal matrix if hij = 0 for all pairs (i; j) such that ji − jj > 1. On the other hand, a matrix H = (hij)n×n is called a lower (or an upper, respectively) Hessenberg matrix if hij = 0 for all pairs (i; j) such that i + 1 < j (or j + 1 < i, respectively). See the paper [22] and closely-related references therein.
    [Show full text]
  • Note on Ward-Horadam H (X)-Binomials' Recurrences And
    Note on Ward-Horadam H(x)- binomials' recurrences and related interpretations, II Andrzej Krzysztof Kwa´sniewski Member of the Institute of Combinatorics and its Applications, Winnipeg, Manitoba, Canada PL-15-674 Bia lystok, Konwaliowa 11/11, Poland e-mail: [email protected] Summary This note is a continuation of [48, 2010]. Firstly, we propose H(x)-binomials' recurrence formula appointed by Ward-Horadam H(x) = hHn(x)in≥0 functions' sequence i.e. any functions' sequence solution of the second order recurrence with functions' coefficients. As a method this comprises H ≡ H(x = 1) num- ber sequences V -binomials' recurrence formula determined by the primordial Lucas sequence of the second kind V = hVnin≥0 as well as its well elaborated companion fundamental Lucas sequence of the first kind U = hUnin≥0 which gives rise in its turn to the known U-binomials' recurrence as in [1, 1878] , [6, 1949], [8, 1964], [12, 1969], [17, 1989] or in [18, 1989] etc. Then we deliver a new type H(x)-binomials' "mixed" recurrence, straightforward n n formula for special case of Ward-Horadam sequences H(x) = hp (x) + q (x)in≥0. Here general H(x)-binomials' array is appointed by Ward-Horadam sequence of functions which in predominantly considered cases where chosen to be polyno- mials. For the sake of combinatorial interpretations and in number theory H(x = 1) = hHn(x = 1)in≥0 is usually considered to be natural or integer numbers valued sequence. Number sequences H = H(x = 1) = hHnin≥0 were also recently called by several authors: Horadam sequences.
    [Show full text]
  • Various Old and New Results in Classical Arithmetic by Special Functions
    VARIOUS OLD AND NEW RESULTS IN CLASSICAL ARITHMETIC BY SPECIAL FUNCTIONS A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science By Michael Henry May 2018 c Copyright All rights reserved Except for previously published materials Thesis written by Michael Andrew Henry B.A., The University of Maryland College Park, 2009 M.S., Kent State University 2018 Approved by Gang Yu _____________________________, Advisor Andrew Tonge_________________________, Chair, Department of Mathematical Sciences James L. Blank_________________________, Dean, College of Arts and Sciences Table of Contents Page List of Tables . iv Acknowledgments . v 1. INTRODUCTION . 1 2. BASIC NUMBER THEORY RESULTS . 2 3. IRRATIONALITY . 5 4. TRANSCENDENCE . 14 5. ELEMENTARY CONTINUED FRACTIONS . 20 6. LAMBERT SERIES, #-FUNCTIONS, AND LANDAU’S RECIPROCAL FIBONACCI SUM............................................ 30 7. FIBONACCI POLYNOMIALS AND MORE LANDAU SUMS . 39 8. LATERAL LANDAU SUMS . 44 9. SPECULATIONS AND FINAL REMARKS . 47 iii List of Tables Table Page 5.1 Continuant polynomials . 22 7.1 Fibonacci polynomials . 42 7.2 Values of qn(k) .................................... 43 8.1 A glimpse of the spectrum . 46 iv Acknowledgments I would like to thank Dr. Gang Yu for his assistance in a reading course in number theory that was the basis of this work; I am very grateful for the degree of independence that he has given me throughout my studies. I would like to thank Dr. Ulrike Vorhauer for her encouragement and enthusiasm for me to continue. Furthermore, she has a clarity of thought that is enviable and that has benefited the final form of this thesis tremendously.
    [Show full text]
  • Ivica Martinjak Faculty of Science, University of Zagreb, Zagreb, Croatia [email protected]
    #A2 INTEGERS 19 (2019) COMPLEMENTARY FAMILIES OF THE FIBONACCI-LUCAS RELATIONS Ivica Martinjak Faculty of Science, University of Zagreb, Zagreb, Croatia [email protected] Helmut Prodinger Department of Mathematics, University of Stellenbosch, Stellenbosch, South Africa [email protected] Received: 5/11/18, Revised: 10/8/18, Accepted: 12/15/18, Published: 1/5/19 Abstract In this paper we present two families of Fibonacci-Lucas identities, with the Sury’s identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci and Lucas sequences we also provide a bijective proof. Both families are then treated by generating functions. 1. Introduction The Fibonacci and Lucas sequences of numbers are defined as Fn+2 = Fn+1 + Fn, F0 = 0, F1 = 1, (1) Ln+2 = Ln+1 + Ln, L0 = 2, L1 = 1, (2) where n N0, and denoted by (Fn)n 0 and (Ln)n 0, respectively. Equivalently, 2 ≥ ≥ these sequences could be defined as the only solutions (x, y), x = Ln, y = Fn of the Diophantine equation x2 5y2 = 4( 1)n. − − For natural numbers k, n we consider the number of ways to tile a k n rectangle ⇥ with certain tiles. The area of a 1 1 rectangle is usually called a cell. Tiles that are ⇥ 1 1 rectangles are called squares and tiles of dimension 1 2 are called dominoes. ⇥ ⇥ In particular, when k = 1 a rectangle to tile is called n-board. Among many interpretations of the Fibonacci and Lucas numbers, here we use the fact that the number fn of n-board tilings with squares and dominoes is equal to Fn+1, while the number ln of the same type of tilings of a labeled circular n-board INTEGERS: 19 (2019) 2 (labeled in the clockwise direction) is equal to Ln.
    [Show full text]
  • Elliptic and $ Q $-Analogs of the Fibonomial Numbers
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 076, 16 pages Elliptic and q-Analogs of the Fibonomial Numbers Nantel BERGERON y, Cesar CEBALLOS z and Josef KUSTNER¨ x y Department of Mathematics and Statistics, York University, Toronto, Canada E-mail: [email protected] URL: http://www.math.yorku.ca/bergeron/ z Institute of Geometry, TU Graz, Graz, Austria E-mail: [email protected] URL: http://www.geometrie.tugraz.at/ceballos/ x Faculty of Mathematics, University of Vienna, Vienna, Austria E-mail: [email protected] URL: http://homepage.univie.ac.at/josef.kuestner/ Received March 14, 2020, in final form July 29, 2020; Published online August 13, 2020 https://doi.org/10.3842/SIGMA.2020.076 Abstract. In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial de- scription for the q-analog and elliptic analog of the Fibonomial numbers. This is achieved by introducing some q-weights and elliptic weights to a slight modification of the combinatorial model of Sagan and Savage. Key words: Fibonomial; Fibonacci; q-analog; elliptic analog; weighted enumeration 2020 Mathematics Subject Classification: 11B39; 05A30; 05A10 1 Introduction The Fibonacci sequence 0; 1; 1; 2; 3; 5; 8; 13; 21; 34;::: is one of the most important and beautiful sequences in mathematics. It starts with the numbers F0 = 0 and F1 = 1, and is recursively defined by the formula Fn = Fn−1 + Fn−2.
    [Show full text]
  • A Two Variable Lucas Polynomials Corresponding to Hybrid Fibonacci Polynomials
    Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1669–1682 © Research India Publications http://www.ripublication.com/gjpam.htm A Two Variable Lucas Polynomials Corresponding to Hybrid Fibonacci Polynomials R. Rangarajan Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru - 570 006, INDIA. Honnegowda C.K.1 Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru - 570 006, INDIA. Shashikala P. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru - 570 006, INDIA. Abstract Fibonacci numbers, Fibonacci polynomials in one variable, namely Catalan poly- nomials and Jacobsthal polynomials are recently generalized by the authors to two variables in the form of hybrid Fibonacci polynomials which exhibit many interest- ing combinatorial properties useful for research workers in combinatorics. In the present paper, two variable Lucas polynomials corresponding to two variable hybrid Fibonacci polynomials are defined and studied their combinatorial properties. AMS subject classification: 05A19, 30B70 and 33C70. Keywords: Combinatorial identities, Fibonacci and Lucas numbers and hyper ge- ometric functions in one or more variables. 1Corresponding author. 1670 R. Rangarajan, Honnegowda C.K., and Shashikala P. 1. Introduction (C) (J ) Two Lucas polynomials denoted ln (x) and ln (x) are defined in the literature corre- (C) sponding to Catalan polynomials and Jacobsthal polynomials denoted by fn (x) and (J ) fn (x) respectively which are natural extension of Fibonacci numbers Fn to one variable polynomials with many interesting combinatorial properties relevant to current literature [5, 7, 8, 9, 11, 12, 15]. The natural and the most beautiful three term recurrence relation is Fn+1 = Fn + Fn−1,F1 = 1,F2 = 1,n= 1, 2, 3,...[1, 3, 4, 16] given by Fibonacci numbers.
    [Show full text]
  • 2-Fibonacci Polynomials in the Family of Fibonacci Numbers
    Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 24, 2018, No. 3, 47–55 DOI: 10.7546/nntdm.2018.24.3.47-55 2-Fibonacci polynomials in the family of Fibonacci numbers Engin Özkan1, Merve Ta¸stan1 and Ali Aydogdu˘ 2 1 Department of Mathematics, University of Erzincan Binali Yıldırım Faculty of Arts and Sciences, Yalnızbag˘ Campus, 24100, Erzincan, Turkey e-mails: [email protected], [email protected] 2 Department of Mathematics, University of Beykent Ayazaga˘ Campus, Ayazaga-˘ Maslak, Sarıyer, 34485, Istanbul,˙ Turkey e-mail: [email protected] Received: 3 April 2018 Revised: 17 August 2018 Accepted: 19 August 2018 Abstract: In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4]. We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal’s triangle. We give some identities of the 2-Fibonacci polynomials. Afterwards, we compare the polynomials with known Fibonacci poly- nomials. We also express 2-Fibonacci polynomials by the Fibonacci polynomials. Furthermore, we prove some theorems related to the polynomials. Also, we introduce the derivative of the 2-Fibonacci polynomials. Keywords: Fibonacci numbers, Fibonacci polynomials, Generalized Fibonacci polynomials. 2010 Mathematics Subject Classification: 11B39. 1 Introduction Fibonacci numbers are of a great importance in mathematics and they have many important ap- plications to diverse fields (see, e.g., [7]). The Fibonacci numbers Fn are given by the recurrence relation Fn+1 = Fn + Fn−1 ; n ≥ 1 47 with the initial conditions F0 = 0 and F1 = 1.
    [Show full text]