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Chaos, Solitons and Fractals 39 (2009) 1005–1019 www.elsevier.com/locate/chaos On k-Fibonacci sequences and polynomials and their derivatives Sergio Falco´n *,A´ ngel Plaza Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), 35017 Las Palmas de Gran Canaria, Spain Accepted 30 March 2007 Abstract The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of con- volution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction There is a huge interest of modern science in the application of the Golden Section and Fibonacci numbers [1–10]. The Fibonacci numbers Fn are the terms of the sequence f0; 1; 1; 2; 3; 5; ...g wherein each term is the sum of the two previous terms, beginning with the values F0 = 0, and F1 = 1. On thep otherffiffi hand the ratio of two consecutive Fibonacci 1þ 5 numbers converges to the Golden Mean, or Golden Section, / ¼ 2 , which appears in modern research, particularly physics of the high energy particles [11,12] or theoretical physics [13–19]. The paper presented here was initially originated for the astonishing presence of the Golden Section in a recursive partition of triangles in the context of the finite element method and triangular refinements. In [20] we showed the rela- tion between the 4-triangle longest-edge partition and the k-Fibonacci numbers, as another example of the relation between geometry and numbers. On the other hand in [21] the k-Fibonacci numbers were given in an explicit way and, by easy arguments, many properties were proven. In particular the k-Fibonacci numbers were related with the so-called Pascal 2-triangle. From this point on, the present paper is organized as follows. In Section 2 a brief summary of the previous results obtained in References [20–22] is given. Section 3 is focused on the k-Fibonacci polynomials which are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here, in particular we relate the Catalan’s numbers with the coefficients of the successive powers xn written as linear combination of Fibo- nacci polynomials. Section 4 presents many formulas and relations for the derivatives of the Fibonacci polynomials. For example, their derivatives are given as convolution of Fibonacci polynomials. This fact allows us to present a family of integer sequences in a new and direct way. * Corresponding author. Tel.: +34 928 45 88 27; fax: +34 928 45 88 11. E-mail address: [email protected] (S. Falco´n). 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.03.007 1006 S. Falco´n, A´ . Plaza / Chaos, Solitons and Fractals 39 (2009) 1005–1019 2. The k-Fibonacci numbers and properties The k-Fibonacci numbers have been defined in [21] for any real number k as follows. Definition 1. For any positive real number k, the k-Fibonacci sequence, say fF k;ngn2N is defined recurrently by F k;nþ1 ¼ kF k;n þ F k;nÀ1 for n P 1: ð1Þ with initial conditions F k;0 ¼ 0; F k;1 ¼ 1 ð2Þ Note that if k is a real variable x then F k;n ¼ F x;n and they correspond to the Fibonacci polynomials defined by 8 9 <> 1ifn ¼ 0 => F ðxÞ¼ x if n ¼ 1 nþ1 :> ;> xF nðxÞþF nÀ1ðxÞ if n > 1 Particular cases of the k-Fibonacci sequence are • If k = 1, the classical Fibonacci sequence is obtained: F 0 ¼ 0; F 1 ¼ 1; and F nþ1 ¼ F n þ F nÀ1 for n P 1 : fF ngn2N ¼f0; 1; 1; 2; 3; 5; 8; ...g • If k = 2, the Pell sequence appears: P 0 ¼ 0; P 1 ¼ 1; and P nþ1 ¼ 2P n þ P nÀ1 for n P 1 : fP ngn2N ¼f0; 1; 2; 5; 12; 29; 70; ...g • If k = 3, the following sequence appears: H 0 ¼ 0; H 1 ¼ 1; and H nþ1 ¼ 3H n þ H nÀ1 for n P 1 : fH ngn2N ¼f0; 1; 3; 10; 33; 109; ...g The well-known Binet’s formula in the Fibonacci numbers theory [1,8,21] allows us to express the k-Fibonacci number in function of the roots r1 and r2 of the characteristic equation, associated to the recurrence relation (1) r2 = kr +1: n n r1 À r2 F k;n ¼ ð3Þ r1 À r2 rnÀrÀn If r denotes the positive root of the characteristic equation, the general term may be written in the form F k;n ¼ rþrÀ1 , and the limit of the quotient of two terms is F lim k;nþr ¼ rr ð4Þ n!1 F k;n In addition, the general term of the k-Fibonacci sequence may be obtained by the formula: bcnÀ1 1 X2 n nÀ2iÀ1 2 i ð5Þ F k;n ¼ nÀ1 k ðk þ 4Þ 2 i¼0 2i þ 1 or, equivalently, by nÀ1 bcX2 n À 1 À i nÀ1À2i F k;n ¼ k ð6Þ i¼0 i S. Falco´n, A´ . Plaza / Chaos, Solitons and Fractals 39 (2009) 1005–1019 1007 Table 1 The first k-Fibonacci numbers F k;1 ¼ 1 F k;2 ¼ k 2 F k;3 ¼ k þ 1 3 F k;4 ¼ k þ 2k 4 2 F k;5 ¼ k þ 3k þ 1 5 3 F k;6 ¼ k þ 4k þ 3k 6 4 2 F k;7 ¼ k þ 5k þ 6k þ 1 7 5 3 F k;8 ¼ k þ 6k þ 10k þ 4k 2.1. The k-Fibonacci numbers and the Pascal 2-triangle From the definition of the k-Fibonacci numbers, the first of them are presented in Table 1. From these expressions, one may deduce the value of any k-Fibonacci number by simple substitution on the corresponding F k;n. For example, 6 4 2 the seventh element of the 4-Fibonacci sequence, fF 4;ngn2N ,isF 4;7 ¼ 4 þ 5 Á 4 þ 6 Á 4 þ 1 ¼ 5473. By doing k ¼ 1; 2; 3; ... the respective k-Fibonacci sequences are obtained whose first elements are fF 1;ngn2N ¼f0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55 ...g fF 2;ngn2N ¼f0; 1; 2; 5; 12; 29; 70; 169; 408; 985; 2378 ...g fF 3;ngn2N ¼f0; 1; 3; 10; 33; 109; 360; 1189; 3927; 12970; 42837; ...g fF 4;ngn2N ¼f0; 1; 4; 17; 72; 305; 1292; 5473; 23184; 98209; 416020; ...g Sequence fF 1;ng is the classical Fibonacci sequence and fF 2;ng is the Pell sequence. It is worthy to be noted that only the first 11 k-Fibonacci sequences are referenced in The On-Line Encyclopedia of Integer Sequences [23] with the numbers given in Table 2. For k even with 12 6 k 6 62 sequences {Fk,n} are referenced without the first term Fk,0 =0in[23]. It is worthy to be noted that the coefficients arising in the previous list, see Table 1, can be written in triangular posi- tion, in such a way that every side of the triangle is double, and for this reason this triangle has been called Pascal 2- triangle [21]. See Table 3. Table 2 The first 11 k-Fibonacci sequences as numbered in The On-Line Encyclopedia of Integer Sequences [23] fF 1;ng A000045 fF 7;ng A054413 fF 2;ng A000129 fF 8;ng A041025 fF 3;ng A006190 fF 9;ng A099371 fF 4;ng A001076 fF 10;ng A041041 fF 5;ng A052918 fF 11;ng A049666 fF 6;ng A005668 Table 3 The Pascal 2-triangle 1. 1 2. 1 3. 1 1 4. 1 2 5. 1 3 1 6. 1 4 3 7. 1561 8. 1 6 10 4 9. 1715101 10. 1821205 11.192835151 12.1103656356 13.1114584 70211 14. 1 12 55 120 126 56 7 1008 S. Falco´n, A´ . Plaza / Chaos, Solitons and Fractals 39 (2009) 1005–1019 Table 4 New table from the Classical Pascal triangle 1111111... 1234567... 13610152128... 1 4 10 20 35 56 84 ... 1 5 15 35 70 126 210 ... 1 6 21 56 126 252 462 ... ... Note that the numbers belonging to the same row of the Pascal 2-triangle are the coefficients of F k;n as they are expressed in Eq. (6). Also the elements belonging to the odd rows in inverse way build the Modified Numerical Triangle (MNT) introduced by Trzaska in [24–26], and defined by the recurrence equation T nþ2ðxÞ¼ð2 þ xÞT nþ1ðxÞT nðxÞ, for n ¼ 0; 1; 2; ..., with initial values T0(x) = 1 and T1(x)=1+x. On the other hand, the even rows, written in inverse way, form the so-called (MNT2) triangle introduced by Trzaska in [26], and defined by the recurrence equation P nþ2ðxÞ¼ð2 þ xÞP nþ1ðxÞP nðxÞ, for n ¼ 0; 1; 2; ..., with initial values P0(x) = 1 and P1(x)=1. A simple explanation of the Pascal 2-triangle may be given by considering two sets of points in the coordinate axes X ¼fx ¼ðx; 0Þ=x 2 Ng and Y ¼fy ¼ð0; yÞ=y 2 Ng. A path between an x-point and a y-point is the not reversing path in the first quadrant from x to y by horizontal and vertical unit segments.
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  • Geometry in Design Geometrical Construction in 3D Forms by Prof

    Geometry in Design Geometrical Construction in 3D Forms by Prof

    D’source 1 Digital Learning Environment for Design - www.dsource.in Design Course Geometry in Design Geometrical Construction in 3D Forms by Prof. Ravi Mokashi Punekar and Prof. Avinash Shide DoD, IIT Guwahati Source: http://www.dsource.in/course/geometry-design 1. Introduction 2. Golden Ratio 3. Polygon - Classification - 2D 4. Concepts - 3 Dimensional 5. Family of 3 Dimensional 6. References 7. Contact Details D’source 2 Digital Learning Environment for Design - www.dsource.in Design Course Introduction Geometry in Design Geometrical Construction in 3D Forms Geometry is a science that deals with the study of inherent properties of form and space through examining and by understanding relationships of lines, surfaces and solids. These relationships are of several kinds and are seen in Prof. Ravi Mokashi Punekar and forms both natural and man-made. The relationships amongst pure geometric forms possess special properties Prof. Avinash Shide or a certain geometric order by virtue of the inherent configuration of elements that results in various forms DoD, IIT Guwahati of symmetry, proportional systems etc. These configurations have properties that hold irrespective of scale or medium used to express them and can also be arranged in a hierarchy from the totally regular to the amorphous where formal characteristics are lost. The objectives of this course are to study these inherent properties of form and space through understanding relationships of lines, surfaces and solids. This course will enable understanding basic geometric relationships, Source: both 2D and 3D, through a process of exploration and analysis. Concepts are supported with 3Dim visualization http://www.dsource.in/course/geometry-design/in- of models to understand the construction of the family of geometric forms and space interrelationships.