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2 Available online at Chaos, Solitons and Fractals 38 (008) The -Fibonacci hyperbolic functions Sergio Falcón *,Ángel Plaza Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), Las Palmas de Gran Canaria, Spain Accepted 10 November 006 Abstract An extension of the classical hyperbolic functions is introduced and studied. These new -Fibonacci hyperbolic functions generalize also the -Fibonacci sequences, say ff ;n g 1 n¼0, recently found by studying the recursive application of two geometrical transformations onto C ¼ C [ fþ1g used in the well-nown four-triangle longest-edge (4TLE) partition. In this paper, several properties of these -Fibonacci hyperbolic functions are studied in an easy way. We finalize with the introduction of some curves and surfaces naturally related with the -Fibonacci hyperbolic functions. Ó 006 Elsevier Ltd. All rights reserved. 1. Introduction One of the simplest and most celebrated integer sequences is the Fibonacci sequence. The Fibonacci sequence is {F n } = {0,1,1,,3,5,...} wherein each term is the sum of the two preceding terms, beginning with the values F 0 =0, and F 1 = 1. It is interesting to emphasize the fact that the ratio of two consecutive Fibonacci numbers converges to the Golden Mean, or Golden Section, / ¼ ffiffi 1þ p 5, which appears in modern research in many fields from Architecture [1,] to physics of the high energy particles [3 6] or theoretical physics [7 13]. Golden Section and Fibonacci numbers have been studied from different points of view in modern science [14 4].In a recent paper [5] we showed the relation between the four-triangle longest-edge (4TLE) partition and the -Fibonacci numbers, as another example of the relation between geometry and numerical sequences. Moreover, the -Fibonacci numbers have been exposed in an explicit way and are related with the so-called Pascal -triangle [6]. This paper presents the continuous versions of the -Fibonacci numbers which are the -Fibonacci functions. These functions arise naturally from the -Fibonacci numbers and can be seen as a new type of hyperbolic functions, where the golden ratio / = r 1, or more generally r for each real number, plays an analogous role that the number e into the classical hyperbolic functions The -Fibonacci numbers For any positive real number, the -Fibonacci sequence, say {F,n } nn is defined recurrently by [5]: F ;0 ¼ 0; F ;1 ¼ 1; F ;nþ1 ¼ F ;n þ F ;n 1 for n P 1 * Corresponding author. Tel.: ; fax: address: sfalcon@dma.ulpgc.es (S. Falcón) /$ - see front matter Ó 006 Elsevier Ltd. All rights reserved. doi: /j.chaos

3 410 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) Particular cases of the -Fibonacci sequence for = 1, are, respectively, the classical Fibonacci sequence: F 0 =0,F 1 =1,F n+1 = F n + F n 1 for n P 1, and the Pell sequence: P 0 =0,P 1 =1,P n+1 =P n + P n 1 for n P 1. Note that the characteristic equation associated with the recursive definition of the -Fibonacci numbers is r ¼ r þ 1 ð1þ Let us consider the geometric sequence {1,r,r,...,r n,...} in which r is the positive root of Eq. (1), that is r = Ær + 1. This geometric sequence is a particular case of a -Fibonacci sequence, say {F,n } nn, in which F,0 =1,F,1 = r, F,n+1 = F,n + F,n 1 for n P 1. It should be noted that every geometric sequence can be seen as a -Fibonacci sequence, for the appropriate value of, since and r are related by r = Ær + 1. Note that in this sequence, if r 5 1 the sequence is non-trivial and, as all geometric sequence, verifies that the quotient of two consecutive terms is equal to a constant number called the ratio of the sequence. We will call here the previous sequence by metallic sequence, and this name will be justified in the sequel. One of the most common formulas in the context of the Fibonacci theory is Binet s formula, which for the -Fibonacci numbers are given by (see [5]): F ;n ¼ rn 1 rn r 1 r where r 1, r are the roots of the characteristic equation (1) defining F,n. An useful property in these sequences is that the limit of the quotient of two consecutive terms is equal to the positive F root of the corresponding characteristic equation (see [5]): lim ;n n!1 F ;n 1 ¼ r 1 where r 1 is the positive root of Eq. (1).. New -Fibonacci hyperbolic functions As is well-nown, the classical hyperbolic functions are defined as follows: cosh x ¼ ex þ e x sinh x ¼ ex e x On the other hand, recently, the Fibonacci hyperbolic functions have been defined as [19,1,3]: sfhx ¼ /x / x pffiffi ¼ wx w x pffiffi 5 5 cfhx ¼ /ðxþ1þ þ / ðxþ1þ pffiffi ¼ wxþ1= þ w ðxþ1=þ pffiffi 5 5 where sfh and cfh are called, respectively, the Fibonacci hyperbolic sine and cosine, / ¼ ffiffi 1þ p 5, and w = / =1+/. The above functions may be extended to the -Fibonacci hyperbolic functions in the following way: sf hðxþ ¼ rx r x þ 4 cf hðxþ ¼ rðxþ1þ þ r ðxþ1þ þ 4 under the condition pffiffiffiffiffiffiffithat r is the positive root of the characteristic equation associated to the -Fibonacci sequence, that is r ¼ þ þ4. Notice that these functions verify the property that, if x is an even number, x =n then sf h(x)=f,n, while if x is an odd number, x =n + 1 then cf h(x)=f,n+1. Note that sf h(x) is symmetric with respect to the origin, while the graphic of cf h(x) presents a symmetry with respect to the axis x ¼ 1. By doing a change of variable x +1=t in function cf h, the symmetry is for the new variable t with respect to the axis t =0. For this reason, from now on, in analogous way followed by Stahov and Rozin [19,1] the so-called -Fibonacci hyperbolic sine and cosine functions are, respectively, defined as follows: r x sf hðxþ ¼ rx r þ r 1 cf hðxþ ¼ rx þ r x r þ r 1

4 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) a b Fig. 1. The Fibonacci hyperbolic functions sfh(x) (left) and cfh(x) (right). since r þ r 1 ¼ þ 4. The graphics of these new -Fibonacci hyperbolic functions are shown in Fig. 1 with the more appealing symmetry with respect to the vertical axis x = 0 for the cosine function, and, besides cf hð0þ ¼p ffiffiffiffiffiffiffi,cf h(1) = 1. In addition, also by the last definitions, it holds that sf h(n)=f,n and cf h(n +1)=F,n+1, as it can be easily n verified by using r n ¼ 1 ðnþ1þ nþ1. r and r ¼ 1 r The -Fibonacci hyperbolic functions are related with the classical hyperbolic functions by the following identities: sf hðxþ ¼ sinhðx ln r r þ r 1 Þ cf hðxþ ¼ coshðx ln r r þ r 1 Þ From now on for simplicity we will write r instead of r..1. Properties of the -Fibonacci hyperbolic functions In the sequel we present the main properties of these functions in a similar way in which the analogous properties of the classical hyperbolic functions are usually presented. Proposition 1 (Pythagorean theorem). The main property of these functions which it may be called as a version of the Pythagorean Theorem is ½cF hðxþš ½sF hðxþš ¼ 4 ðþ þ 4 Notice, that since 5 0 in the previous identity the right hand side is a positive number less than one. Proposition (Sum and difference). þ 4 cf hðx þ yþ ¼ ðcf hðxþcf hðyþþsf hðxþsf hðyþþ þ 4 cf hðx yþ ¼ ðcf hðxþcf hðyþ sf hðxþsf hðyþþ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 sf hðx þ yþ ¼ ðsf hðxþcf hðyþþcf hðxþsf hðyþþ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 sf hðx yþ ¼ ðsf hðxþcf hðyþ cf hðxþsf hðyþþ Proof. Let us prove the first identity: þ4

5 41 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) cf hðxþcf hðyþþsf hðxþsf hðxþ ¼ ðrx þ r x Þðr y þ r y Þþðr x r x Þðr y r y Þ ð Þ ¼ r þ r rxþy þ r ðxþyþ ¼ cf 1 hðx þ yþ þ 4 from where the first identity is deduced. h By doing y = x in the first and third previous formula, we have the following corollary. Corollary 3 (Double argument). þ 4 cf hðxþ ¼ ð½cf hðxþš þ½sf hðxþš Þ sf hðxþ ¼ þ 4sF hðxþcf hðxþ From Eqs. () and (4) it is deduced, respectively, by summing up and subtracting: ð3þ ð4þ Corollary 4 (Half argument).! ½cF hðxþš 1 ¼ cf hðxþþ þ 4 þ 4! ½sF hðxþš 1 ¼ cf hðxþ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 þ 4 Finally, the nth derivatives of these functions verify: Corollary 5 (nth derivatives). ½cF hðxþš ðnþ ¼ ðln rþn sf hðxþ for n ¼ m þ 1 ðln rþ n cf hðxþ for n ¼ m ½sF hðxþš ðnþ ¼ ðln rþn cf hðxþ for n ¼ m þ 1 ðln rþ n sf hðxþ for n ¼ m It should be noted that in the case = 0 the -Fibonacci sequence would be the sequence {0,1,0,1,0,1,...} with r =1. In addition, by doing = 0 in the previous formulas the expressions for the classical hyperbolic functions arise. For example, the Pythagorean Theorem (or fundamental identity) taes the well-nown form [cosh(x)] [sinh(x)] =1 and the identities for the half argument are ½coshðxÞŠ ¼ coshðxþþ1, ½sinhðxÞŠ ¼ coshðxþ 1. Therefore these -Fibonacci hyperbolic functions generalize the classical hyperbolic functions, for a general value R... The -Fibonacci hyperbolic functions and the -Fibonacci numbers The -Fibonacci hyperbolic functions are related with the -Fibonacci numbers, and hence, this relation appears in many expressions. Some of these identities are presented here. By definition, the -Fibonacci numbers verify that F,0 =0,F,1 =1,F,n+1 = F,n + F,n 1. An analogous equation for the -Fibonacci hyperbolic functions is the following: Proposition 6 (Recursive relation). sf hðx þ 1Þ ¼ cf hðxþþsf hðx 1Þ Proof. Since r = r + 1, then r x + r x 1 = r x 1 (r +1)=r x+1. In addition, r x r x 1 ¼ r r x Therefore cf hðxþþsf hðx 1Þ ¼ rxþ1 r ðxþ1þ ¼ sf rþr 1 hðx þ 1Þ. h Catalan s identity for the -Fibonacci numbers [5] is F ;n r F ;nþr F ;n ¼ð 1Þnþ1 r F ;r ¼ r r r xþ1 ¼ 1 r xþ1.

6 and for the -Fibonacci hyperbolic functions results: Proposition 7 (Catalan s identity). cf hðx rþcf hðx þ rþ ½cF hðxþš ¼½sF hðrþš Proof. By definition, the left hand side (LHS) is ðlhsþ ¼ ðrx r þ r xþr Þðr xþr þ r x r Þ ðr x þ r x Þ ð Þ ¼ ðrr r r Þ ð Þ ¼½sF hðrþš In a similar way it can be obtained: Corollary 8. cf hðx rþcf hðx þ rþ ½sF hðxþš ¼½cF hðrþš sf hðx rþsf hðx þ rþ ½sF hðxþš ¼ ½sF hðrþš sf hðx rþsf hðx þ rþ ½cF hðxþš ¼ ½cF hðrþš ¼ rx þ r r þ r r þ r x r x r x ð Þ By doing r = 1 into classical Catalan s identity it is straightforwardly obtained for the -Fibonacci numbers Cassini s or Simson s identity [5]: F ;n 1 F ;nþ1 F ;n ¼ð 1Þn The corresponding identity for the -Fibonacci hyperbolic functions is Proposition 9 (Cassini s or Simson s identity). S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) cf hðx 1ÞcF hðx þ 1Þ ½sF hðxþš ¼ 1 sf hðx 1ÞsF hðx þ 1Þ ½cF hðxþš ¼ 1 d Ocagne s identity for the -Fibonacci numbers [5] is F ;m F ;nþ1 F ;mþ1 F ;n ¼ð 1Þ n F ;m n if m > n which is generalized for the -Fibonacci hyperbolic functions as: Proposition 10 (d Ocagne s identity). cf hðxþcf hðy þ rþ sf hðx þ rþsf hðyþ ¼cF hðrþcf hðx yþ cf hðxþsf hðy þ rþ cf hðx þ rþsf hðyþ ¼sF hðrþcf hðx yþ ð5þ ð6þ Proof. Noting by (LHS) the left hand side of Eq. (5) we have: ðlhsþ ¼ ðrx þ r x Þðr yþr þ r y r Þ ðr xþr r x r Þðr y r y Þ ð Þ ¼ r r ðr x y þ r ðx yþ Þþr r ðr x y þ r ðx yþ Þ ð Þ ¼ rx y r þ r xþyþr þ r x yþr þ r xþy r ð Þ ¼ rr þ r r r þ r rx y þ r ðx yþ ¼ cf 1 hðrþcf hðx yþ Eq. (6) can be deduced similarly. h Notice that by doing r = 1 in Eq. (5) it is obtained: Corollary 11. cf hðxþcf hðy þ 1Þ sf hðx þ 1ÞsF hðyþ ¼cF hðx yþ

7 414 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) The quasi-sine -Fibonacci function Binet s formula for the -Fibonacci sequence establishes that the general term of this sequence can be written as F ;n ¼ rn ð 1Þ n r n, where r = r rþr 1 is the positive root of the characteristic equation [5]. On the other hand, we have defined here the -Fibonacci hyperbolic sine function as sf hðxþ ¼ rx r x. Therefore, function sf rþr 1 h(x), if x is an even integer number, taes the value corresponding to the -Fibonacci sequence, that is sf h(x) =F,n. Therefore, and taing into account that cos(np) =( 1) n it is natural to introduce the following definition. Definition 1 [3]. The quasi-sine -Fibonacci function is defined by FF ðxþ ¼ rx cosðpxþr x Notice that FF (n)=f,n for all integer n, so the points of the form (n,f,n ) belong to the graphics of these functions. Fig. shows the graphics of FF (x) for = 1,,3. Note that although the general shape of these functions is quite similar, however they differ in many details as the values of their relative extremes and the number of such extremes. For example, for positive values of x it is observed that FF 1 (x) presents an unique maximum at x = and an unique minimum at x = while for values of > 1 function FF (x) does not have relative extremes for x > 0. In addition the number of relative extremes for x >0 decreases when increases. For example, for = 0.8 there is a unique maximum and a unique minimum; however, for = 0.5 there are two maximums and two minimums; and for = 0.5 there are six maximums and six minimums; and so on. As a matter of example, Fig. 3 shows the graphic of FF (x) for = Many properties of these functions can be investigated, even from an experimental point of view. For instance, for = 1, then FF 1 (x)=f n for any integer x = n, and since F 1 = F = 1 and F m > F n for m > n >, function FF 1 (x) presents one maximum and one minimum into the interval (1,), and no more extremes for x >. However, for N {1,}, function FF (x) increases for x P 0 and, hence, it has not any extreme in [0, + 1). Therefore, by continuity, there should be some value of, say 0, between 1 and such that for any > 0, function FF (x) has not extremes for positive x. By using MATHEMATICA Ó this value can be approximately calculated: y = FF1 () x y = FF () x a b y = FF3 () x c Fig.. The quasi-sine -Fibonacci functions, FF (x), for =1,,3.

8 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) y = FF0.01() x Fig. 3. The quasi-sine 0.01-Fibonacci function. Finally, the infinite relative extreme points of these functions for x < 0 are presented at different values of x. For example, the first minimum for x < 0 and = 1 is at x = while for = is at x = , and for =3 is at x = Fig. 4 shows the graphics of FF (x) for = 1, along with their evolving tangent curves which are precisely the - Fibonacci cosine and sine hyperbolic functions. Notice that the tangent points are of the form (n,f,n ) for n Z The quasi-sine -Fibonacci functions and the -Fibonacci numbers There are several identities for these quasi-sine -Fibonacci functions which are versions of analogous identities of the -Fibonacci numbers [19,1,3,5,6]. Here, some of them are exposed. The first result loos lie the recursive relation defining the -Fibonacci numbers. Theorem 13 (Recursive relation). FF ðx þ Þ ¼ FF ðx þ 1ÞþFF ðxþ y = FF1 () x y = FF () x a b Fig. 4. The quasi-sine -Fibonacci functions for = 1, with their evolving tangent curves.

9 416 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) Proof. FF ðx þ 1ÞþFF ðxþ ¼ rxþ1 cosðpðx þ 1ÞÞr x 1 ¼ rxþ1 þ cosðpxþr x 1 þ rx cosðpxþr x þ rx cosðpxþr x ¼ rx ðr þ 1ÞþcosðpxÞr x ðr r Þ ¼ rxþ þ cosðpx þ pþr x ðr r 1Þ ¼ FF ðx þ Þ Similarly to Catalan s identity for the -Fibonacci numbers, for these functions we obtain: Theorem 14 (Catalan s identity). If r Z, then: FF ðx þ rþff ðx rþ ½FF ðxþš ¼ð 1Þ rþ1 cosðpxþ½ff ðrþš Proof. FF ðx þ rþff ðx rþ ½FF ðxþš ¼ rxþr cosðpðx þ rþþr x r ¼ rxþr þð 1Þ rþ1 cosðpxþr x r rx cosðpxþþcos ðpxþr x ð Þ rx r cosðpðx rþþr xþr rx cosðpxþr x rx r þð 1Þ rþ1 cosðpxþr xþr ¼ ð 1Þrþ1 cosðpxþr r þð 1Þ rþ1 cosðpxþr r þ cosðpxþ ð Þ ¼ ð 1Þrþ1 cosðpxþðr r cosðprþr r Þ ð Þ ¼ð 1Þ rþ1 cosðpxþ½ff ðrþš Note that if in the previous result x is an integer, n, then Catalan s identity for the -Fibonacci numbers appears. Also if r = 1 results: FF ðx þ 1ÞFF ðx 1Þ ½FF ðxþš ¼ cosðpxþ which for x = n an integer gives us Cassini s or Simson s identity [5] for the -Fibonacci numbers: F ;nþ1 F ;n 1 F ;n ¼ð 1Þn Theorem 15 (Asymptotic quotient). For any integer r, it is FF ðx þ rþ lim ¼ r r x!1 FF ðxþ Proof. Taing into account that r > 1, then: FF ðx þ rþ r xþr cosðpðx þ rþþr x r r r þð 1Þ rþ1 cosðpxþ 1 r lim ¼ lim ¼ lim xþr ¼ r r x!1 FF ðxþ x!1 r x cosðpxþr x x!1 1 cosðpxþ 1 r x FF In particular, if = 1, then r = / and lim ðxþ1þ x!1 FF ¼ /. ðxþ 4. 3D curves and surfaces for the -Fibonacci hyperbolic functions We shall introduce here several curves and surfaces naturally related with the -Fibonacci functions studied before.

10 Definition 16. The following complex valued function is called the three-dimensional -Fibonacci spiral: CFF ðxþ ¼ rx cosðpxþr x S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) þ i sinðpxþr x where r is the positive root of the characteristic equation of the -Fibonacci sequence: r r 1=0. Note that RðCFF ðxþþ ¼ FF ðxþ, which is the quasi-sine -Fibonacci function. The shape of the three-dimensional -Fibonacci spiral reminds a three-dimensional spiral as shown in Fig. 5. Function CFF (x) can be written also as CFF ðxþ ¼ rx þie ip ð 1 x Þ r x. rþr 1 Some of the properties of the function CFF (x) are the following: If = 1, the three-dimensional Fibonacci spiral introduced by Stahov [1] is obtained. Note that, in this case, CFF (x) can be written also as CFF 1 ðxþ ¼CFFðxÞ ¼ /x þie ip 1 ð x Þ / x p ffiffi, where / is the golden ratio. 5 If x is an integer, the imaginary part of the three-dimensional -Fibonacci spiral is zero, and hence CFF (n) results in Binet s formula for the general term of the -Fibonacci sequence [5]. Finally, as an immediate consequence of the last fact, for every it is CFF (0) = 0 and CFF (1) = 1. By the definition of these functions we will prove the following theorem. Theorem 17 (Recurrence relation). The three-dimensional -Fibonacci spirals verify the following Fibonacci-type recurrence relation: CFF ðx þ Þ ¼ CFF ðx þ 1ÞþCFF ðxþ: Proof. Let us note by (RHS) the right hand side of the identity to prove. Taing into account that r > 1, and since r +1=r,sor ¼ 1, we have: r ðrhsþ ¼ rxþ1 þ ie ip 1 ð x 1Þ r x 1 þ rx þ ie ipð 1 xþ r x ¼ rx ðr þ 1Þþie ipx r x 1 ðe ip þ re ip Þ ¼ rxþ þ ie ipx r x 1 ð þ rþi ¼ rxþ þ ie ipð 1 ðxþþþ r x ¼ CFF ðx þ Þ Since we have shown Catalan s identity for the -Fibonacci hyperbolic functions, and also for the quasi-sine Fibonacci functions, an analogous identity may be demonstrated for the three-dimensional -Fibonacci spiral. Theorem 18 (Catalan s identity). If r is an integer, then the three-dimensional -Fibonacci spiral verifies: CFF ðx rþcff ðx þ rþ ½CFF ðxþš ¼ð 1Þ rþ1 ½CFF ðrþš : Fig. 5. The three-dimensional -Fibonacci spiral, for = 1.

11 418 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) Proof. Let us note by (LHS) the left hand side of the identity to prove. But e ipð1 x ðlhsþ ¼ rx r þ ie ip 1 ð xþrþ r xþr rxþr þ ie ip 1 ð x rþ r x r rx þ ie ip 1 ð x Þ r x Þ ¼ ie ipx, and e ipr =( 1) r,e i p ð prþ ¼ ið 1Þ r since r is an integer. Therefore, ðlhsþ ¼ ðrx r þð 1Þ rþ1 e ipx r xþr Þðr xþr þð 1Þ rþ1 e ipx r x r Þ ðr x e ipx þ e ipx r x Þ ð Þ ¼ rx þð 1Þ rþ1 e ipx r r þð 1Þ rþ1 e ipx r r þ e ipx r x r x þ e ipx e ipx r x ð Þ ¼ð 1Þ rþ1 e ipx ðr r ð 1Þ r þ r r Þ ð Þ ¼ð 1Þ rþ1 ½CFF ðrþš where if r = 1 Simson s identity results: CFF ðx 1ÞCFF ðx þ 1Þ ½CFF ðxþš ¼ð 1Þ rþ1 Using a similar process, it can be proven the following theorem. Theorem 19 (d Ocagne s identity). If r is an integer, then the three-dimensional -Fibonacci spiral verifies: CFF ðx þ rþcff ðx þ 1Þ CFF ðx þ r þ 1ÞCFF ðxþ ¼e ipx CFF ðrþ: If x = r = n previous formula reduces to CFF ðnþcff ðn þ 1Þ CFF ðn þ 1ÞCFF ðnþ ¼ð 1Þ n CFF ðnþ or equivalently F ;n F ;nþ1 F ;nþ1 F ;n ¼ð 1Þ n F ;n! 4.1. The Metallic Shofars Real and imaginary parts of the three-dimensional -Fibonacci spiral are, respectively RðCFF ðxþþ ¼ r x cosðpxþr x ¼ yðxþ, IðCFF rþr 1 ðxþþ ¼ sinðpxþr x ¼ zðxþ. By considering Y axis as real axis, and Z axis as imaginary axis we rþr 1 can write the following system of equations: ( ) yðxþ rx zðxþ ¼ rþr 1 ¼ sin pxr x rþr 1 cos pxr x rþr 1 By summing up the squares of previous expressions and considering y and z as independent variables the following equation is obtained: y rx þ z r x ¼ ð7þ Eq. (7) corresponds to a surface which have been called in the case = 1 the Golden Shofar [1] with equation y p /x ffiffi 5 þ z ¼ 1, where / is the golden ratio. For the case = the Silver Shofar with equation 5/ x y p ux ffiffi 8 þ z ¼ 1 is obtained. Finally, for = 3 results the Bronze Shofar with equation y wx ffiffiffi 8u x p 13 þ z ¼ 1. 13w x Fig. 6 shows the Metallic Shofars for = 1,,3, while Fig. 7 shows the Golden Shofar and its projections on the coordinate planes. Remar 0. Let z as in Eq. (7), then z =(cf h(x) y)(y sf h(x)).

12 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 38 (008) a b c Fig. 6. The Metallic Shofars: (a) Golden Shofar ( = 1); (b) Silver Shofar ( = ); (c) Bronze Shofar ( = 3). Fig. 7. The Golden Shofar and its projections on the coordinate planes. 5. Conclusions New -Fibonacci hyperbolic functions have been introduced and studied here. These new functions are naturally related with the -Fibonacci sequences. Several properties of these functions have been deduced and related with the

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