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The continuous functions for the Fibonacci and Lucas p-numbers
Chaos, Solitons and Fractals 28 (2006) 1014–1025 www.elsevier.com/locate/chaos
The continuous functions for the Fibonacci and Lucas p-numbers Alexey Stakhov *, Boris Rozin International Club of the Golden Section, 6 McCreary Trail, Bolton, ON, Canada L7E 2C8 Accepted 30 August 2005
Abstract The new continuous functions for the Fibonacci and Lucas p-numbers using Binet formulas are introduced. The article is of a fundamental interest for Fibonacci numbers theory and theoretical physics. Ó 2005 Elsevier Ltd. All rights reserved.
1. New results in Fibonacci number theory Modern science, particularly physics [1–13], widely applies the recurring series of the Fibonacci numbers F(n) f0; 1; 1; 2; 3; 5; 8; 13; 21; 34; . . .g
ð1Þ
and the recurring series of the Lucas numbers L(n) f2; 1; 3; 4; 7; 11; 18; 29; 47; . . .g
ð2Þ
which result from application of the following recurrence relations: F ðnÞ ¼ F ðn 1Þ þ F ðn 2Þ for n > 1; F ð0Þ ¼ 0; F ð1Þ ¼ 1; LðnÞ ¼ Lðn 1Þ þ Lðn 2Þ for n > 1; Lð0Þ ¼ 2; Lð1Þ ¼ 1.
ð3Þ ð4Þ ð5Þ ð6Þ
The ratio pffiffi of the adjacent numbers in the Fibonacci series (1) and Lucas series (2) tends toward at the irrational number s ¼ 1þ2 5 called the Golden Proportion (Golden Mean) [14–16]. In the 19th century the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers [14–16]:
*
Corresponding author. E-mail addresses: [email protected] (A. Stakhov), [email protected] (B. Rozin). URLs: http://www.goldenmuseum.com (A. Stakhov), http://www.goldensection.net (B. Rozin).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.158
A. Stakhov, B. Rozin / Chaos, Solitons and Fractals 28 (2006) 1014–1025
1015
sn ð1Þn sn pffiffiffi ; 5 LðnÞ ¼ sn þ ð1Þn sn ; F ðnÞ ¼
ð7Þ ð8Þ
where s is the Golden Proportion and n = 0, ±1, ±2, ±3 . . . In recent years, researchers developed further the theory of the Golden Section and the Fibonacci numbers [17–30]. Hyperbolic Fibonacci and Lucas functions [21,24,26], which are extensions of Binet formulas for a continuous domain, have a strategic importance for the development of both mathematics and theoretical physics. In a previous article [28], the authors presented a new function of the second power called the Golden Shofar. The graph of this function reminds one of a horn, which is used to blow in the Yom Kippur (The Judgment Day). The sinusoidal Fibonacci function [28] is a further development of a continuous approach to the Fibonacci numbers theory begun in a series of papers [21,24,26]. The definition of the function is F ðxÞ ¼
sx cosðpxÞsx pffiffiffi 5
ð9Þ
Fig. 1. The sinusoidal Fibonacci function. Table 1 The extended Fibonacci and Lucas numbers N F(n) F(n) L(n) L(n)
0 0 0 2 2
1 1 1 1 1
2 1 1 3 3
3 2 2 4 4
4 3 3 7 7
5 5 5 11 11
6 8 8 18 18
7 13 13 29 29
8 21 21 47 47
9 34 34 76 76
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The graph of the sinusoidal Fibonacci function given by Eq. (9) is shown in Fig. 1. Note that for the discrete values of the continuous variable x = n = 0, ±1, ±2, ±3, . . . the function (9) reduces to expression (7). That is, in the discrete points x = n = 0, ±1, ±2, ±3 . . . the values of the sinusoidal Fibonacci function (9) coincide with the extended Fibonacci series (Table 1) and, clearly, the extended Fibonacci numbers are inscribed into the graph of the function in Fig. 1. Similarly, we can introduce a sinusoidal Lucas function: LðxÞ ¼ sx þ cosðpxÞsx ;
ð10Þ
that reduces to expression (8) for discrete values of x = n = 0, ±1, ±2, ±3, . . .. It is again clear that the extended Lucas numbers (Table 1) are inscribed to the graph of the function (10). The introduction of generalized Fibonacci and Lucas numbers or Fibonacci and Lucas p-numbers [17,29] is the next new result within the Fibonacci numbers theory. It is proved in [17] that, for a given integer p, p = 1, 2, 3, . . ., the Fibonacci p-numbers Fp(n) are given by the following recurrence formula: F p ðnÞ ¼ F p ðn 1Þ þ F p ðn p 1Þ for n > p
ð11Þ
with the initial conditions: F p ð0Þ ¼ 0;
F p ð1Þ ¼ F p ð2Þ ¼ ¼ F p ðpÞ ¼ 1;
ð12Þ
where n = 0, ±1, ±2, ±3, . . . In article [28], the authors introduced a new class of recurrent numerical sequences, called the Lucas p-numbers, Lp(n), given by the recurrence relation: Lp ðnÞ ¼ Lp ðn 1Þ þ Lp ðn p 1Þ for n > p
ð13Þ
for a given integer p, p = 1, 2, 3, . . ., and with the initial conditions: Lp ð0Þ ¼ p þ 1;
Lp ð1Þ ¼ Lp ð2Þ ¼ ¼ Lp ðpÞ ¼ 1;
ð14Þ
where n = 0, ±1, ±2, ±3, . . . Note that for p = 1 the classic Fibonacci numbers (1) coincide with the Fibonacci p-numbers and the classic Lucas numbers (2) coincide with the Lucas p-numbers. Hence, the introduction of the Fibonacci and Lucas p-numbers considerably develops this area of Fibonaccis research [14–16]. The original research presented in the article Theory of Binet formulas for the Fibonacci and Lucas p-numbers [29] is a new mathematical result for the Fibonacci numbers theory. In essence, for the first time since Binet derived his formulas, we have derived analytical formulas that allow us to express numerical sequences given by Eqs. (11), (12) and (13), (14) using the roots of the golden algebraic equation: xpþ1 ¼ xp þ 1.
ð15Þ
For the case p = 1, the new Binet formulas correspond to the Binet formulas (7), (8) for the classic Fibonacci and Lucas numbers. The purpose of the present article is to develop continuous functions within the Fibonacci number theory which lead to the hyperbolic Fibonacci and Lucas functions [21,24,26], to the function of the ‘‘Golden Shofar’’ [28], and the sinusoidal Fibonacci and Lucas functions (9), (10). The main goal of the article is to extend the results (9) and (10) for the area of the Fibonacci and Lucas p-numbers. That is, to derive formulas for the continuous functions for the Fibonacci and Lucas p-numbers.
2. The continuous functions for the Fibonacci and Lucas 2-numbers 2.1. The continuous function for the Lucas 2-numbers In [29] we defined the Lucas 2-numbers as follows: L2 ðnÞ ¼ L2 ðn 1Þ þ L2 ðn 3Þ; L2 ð0Þ ¼ 3; L2 ð1Þ ¼ L2 ð2Þ ¼ 1.
ð16Þ ð17Þ
For the case p = 2, the golden algebraic equation takes the following form: x3 x2 1 ¼ 0.
ð18Þ
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This equation has three roots: 2 h þ 2h þ 4 x1 ¼ ; 6h
pffiffiffi ! h 1 1 3 h 2 ; x2 ¼ þ i 12 3h 3 2 6 3h pffiffiffi ! h 1 1 3 h 2 x2 ¼ þ þ i ; 2 6 3h 12 3h 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffi 3 where h ¼ ð116 þ 12 93Þ. In [29], we derive the following Binet formula for the Lucas 2-numbers: L2 ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðx3 Þn ; where n = 0, ±1, ±2, ±3, . . .; x1, x2, x3 are the roots of the algebraic equation (18) given by (19). In the interest of clarity and simplicity, we use the following designations: pffiffiffi h 1 1 3 h 2 . c¼ þ ; d¼ 12 3h 3 2 6 3h
ð19Þ
ð20Þ
ð21Þ
We will use the Moivre formula: ðc i dÞn ¼ ðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ ; where h ¼ arccos
c ffi pffiffiffiffiffiffiffiffi c2 þd 2
ð22Þ
.
If we substitute (22) into (19) and then (19) into (20), we will get 2 n h þ 2h þ 4 L2 ðnÞ ¼ þ ðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ þ ðc2 þ d 2 Þn=2 ½cosðnhÞ þ i sinðnhÞ 6h 2 n h þ 2h þ 4 ¼ þ 2ðc2 þ d 2 Þn=2 cosðnhÞ 6h " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n 2 n h þ 2h þ 4 ðh 2Þ h2 þ 2h þ 4 h2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p n ; þ cos arccos ¼ 6h 6h 2 h2 þ 2h þ 4
ð23Þ
where n = 0, ±1, ±2, ±3, . . . is a discrete variable. If we replace the discrete variable n with the continuous variable x in the expression (23), we will get the following continuous function called the continuous function for the Lucas 2-numbers: " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x 2 x h þ 2h þ 4 ðh 2Þ h2 þ 2h þ 4 2h L2 ðxÞ ¼ ð24Þ þ cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x . 6h 6h 2 h2 þ 2h þ 4 The graph of the function (24) is shown in Fig. 2. 2.2. The continuous function for the Fibonacci 2-numbers In [17] we find the following definition of the Fibonacci 2-numbers: F 2 ðnÞ ¼ F 2 ðn 1Þ þ F 2 ðn 3Þ for n > 3; F 2 ð0Þ ¼ 0; F 2 ð1Þ ¼ F 2 ð2Þ ¼ 1. The Binet formula for the Fibonacci 2-numbers has the following form [29]: !n pffiffiffi pffiffiffi 2 n 2hðh þ 2Þ h2 þ 2h þ 4 ððh þ 2Þ þ i 3ðh 2ÞÞh h2 4h þ 4 3ðh 4Þ þ i F 2 ðnÞ ¼ 3 6h 12h 12h ðh þ 8Þ ðh3 þ 8Þ !n pffiffiffi
p ffiffi ffi ðh þ 2Þ i 3ðh 2Þ h h2 4h þ 4 3ðh2 4Þ þi þ ; 3 12h 12h ðh þ 8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ffi 3 where h ¼ ð116 þ 12 93Þ.
ð25Þ ð26Þ
ð27Þ
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Fig. 2. The continuous function for the Lucas 2-numbers.
In addition to (21), we introduce the following designations: pffiffiffi ðh þ 2Þh 3ðh 2Þh ; b ¼ . a¼ ðh3 þ 8Þ ðh3 þ 8Þ Then expression (27) becomes n 2hðh þ 2Þ h2 þ 2h þ 4 þ ða þ ibÞðc idÞn þ ða ibÞðc þ idÞn . F 2 ðnÞ ¼ 3 6h ðh þ 8Þ
ð28Þ
ð29Þ
If we use the Moivre formula (22), then we have following expression for (29): n 2hðh þ 2Þ h2 þ 2h þ 4 þ ða þ i bÞðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ F 2 ðnÞ ¼ 3 6h ðh þ 8Þ þ ða i bÞðc2 þ d 2 Þn=2 ½cosðnhÞ þ i sinðnhÞ n 2hðh þ 2Þ h2 þ 2h þ 4 ¼ 3 þ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ bðiÞ2 sinðnhÞ 6h ðh þ 8Þ n 2hðh þ 2Þ h2 þ 2h þ 4 þ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ þ b sinðnhÞ ¼ 3 6h ðh þ 8Þ n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hðh þ 2Þ h2 þ 2h þ 4 þ 2 a2 þ b2 ðc2 þ d 2 Þn=2 cosðnh gÞ; ¼ 3 6h ðh þ 8Þ
ð30Þ
a ffi where g ¼ arccos pffiffiffiffiffiffiffiffi . a2 þb2 Therefore,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n n 2hðh þ 2Þ h2 þ 2h þ 4 4h h2 2h þ 4 ðh 2Þ h2 þ 2h þ 4 F 2 ðnÞ ¼ 3 þ 6h 6h ðh þ 8Þ h3 þ 8 " ! !# 2h hþ2
cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 h2 þ 2h þ 4 2 h2 2h þ 4
ð31Þ
A. Stakhov, B. Rozin / Chaos, Solitons and Fractals 28 (2006) 1014–1025
1019
Fig. 3. The continuous function for the Fibonacci 2-numbers.
If we replace the discrete variable n with the continuous variable x in Eq. (31), then we get the following continuous function called the continuous function for the Fibonacci 2-numbers: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x x 2hðh þ 2Þ h2 þ 2h þ 4 4h h2 2h þ 4 ðh 2Þ h2 þ 2h þ 4 F 2 ðxÞ ¼ 3 þ 6h 6h ðh þ 8Þ h3 þ 8 " ! !# 2h hþ2 ð32Þ
cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 h2 þ 2h þ 4 2 h2 2h þ 4 Fig. 3 is the graph of function (32). The continuous functions for the Fibonacci and Lucas 2-numbers have recurrent properties that are similar to the Fibonacci and Lucas 2-numbers. Recurrent properties for the Fibonacci and Lucas 2-numbers
Recurrent properties for the continuous functions for the Fibonacci and Lucas 2-numbers
F2(n) = F2(n 1) + F2(n 3) L2(n) = L2(n 1) + L2(n 3)
F2(x) = F2(x 1) + F2(x 3) L2(x) = L2(x 1) + L2(x 3)
3. The continuous functions for the Fibonacci and Lucas 3-numbers If we use the same reasoning for the Fibonacci and Lucas 3-numbers [17,29], we derive the following continuous functions for these numerical series: L3 ðxÞ ¼ 1:38x þ ð0:819Þx cosðpxÞ þ ð0; 9399Þx cosð1:335xÞ; F 3 ðxÞ ¼ 0:3969 1:38x 0:1592 ð0:819Þx cosðpxÞ þ 0:2365 ð0:9399Þx cosð1:335x 2:097Þ.
ð33Þ ð34Þ
Their graphs are shown in Figs. 4 and 5. The continuous functions for the Fibonacci and Lucas 3-numbers have recurrent properties that are similar to the Fibonacci and Lucas 3-numbers.
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Recurrent properties for the Fibonacci and Lucas 3-numbers
Recurrent properties for the continuous functions for the Fibonacci and Lucas 3-numbers
F3(n) = F3(n 1) + F3(n 4) L3(n) = L3(n 1) + L3(n 4)
F3(x) = F3(x 1) + F3(x 4) L3(x) = L3(x 1) + L3(x 4)
Fig. 4. The continuous function for the Lucas 3-numbers.
Fig. 5. The continuous function for the Fibonacci 3-numbers.
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1021
4. The continuous functions for the Fibonacci and Lucas 4-numbers The continuous functions for the Fibonacci and Lucas 4-numbers [17,29] have the following form: " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x x px h4 12h2 þ 144 h 2 h2 þ 12 þ 2 cos cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ; L4 ðxÞ ¼ þ þ2 6 h 3 6h 2 h4 12h2 þ 144 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h 2 px þ g1 þ 2 a22 þ b22 ðc22 þ d 22 Þx=2 cosðh2 x g2 Þ; þ 2 a21 þ b21 cos F 4 ðxÞ ¼ k 1 6 h 3 where h¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 3 ð108 þ 12 69Þ;
k 1 ¼ 0:38095;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 ¼ arccos a1 a21 þ b21 ;
h2 þ 12 h2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 h4 12h2 þ 144
a2 ¼ 0:1191;
b2 ¼ 0:04577;
a1 ¼ 0:07133;
b1 ¼ 0:2063;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 ¼ arccos a2 a22 þ b22 .
5. The continuous functions for the Fibonacci and Lucas p-numbers (a general case) 5.1. The continuous functions for the Lucas p-numbers The Lucas p-numbers (13) and (14) were introduced in [29]. There we find the proof that the following analytical form, which is the Binet formula for the Lucas p-numbers, represents the Lucas p-numbers: Lp ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðx3 Þn þ þ ðxp Þn þ ðxpþ1 Þn ;
ð35Þ
where x1, x2, x3, . . . , xp, xp+1 are the roots of Eq. (15). If p is even, then Eq. (15) has the only the real root x1, which is equal to the Golden p-proportion sp and p/2 pairs of complex conjugate roots. Let us denote the pair of complex conjugate roots by (ct + i Æ dt) and (ct i Æ dt), where t = 1, 2, . . . , p/2. Then Eq. (35) becomes Lp ðnÞ ¼ ðx1 Þn þ ðc1 þ i d 1 Þn þ ðc1 i d 1 Þn þ ðc2 þ i d 2 Þn þ ðc2 i d 2 Þn þ þ ðcp=2 þ i d p=2 Þn þ ðcp=2 i d p=2 Þn . If we use the Moivre formula (22) and simplify, as in (30), we see that 0 1
0
ð36Þ
1
c1 c2 B C B C Lp ðnÞ ¼ ðx1 Þn þ 2ðc21 þ d 21 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2 2 c21 þ d 1 c22 þ d 2 0 1 cp=2 B C þ 2ðc2p=2 þ d 2p=2 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2p=2 þ d 2p=2
ð37Þ
If p is odd, then Eq. (15) has two real roots, x1 and x2, which have opposite signs. In that case, the positive root x1 coincides with the Golden p-proportion sp; the rest of the p 1 roots are the (p 1)/2 pairs of complex conjugate roots. We denote the pair of complex conjugate roots by (ct + i Æ dt) and (ct i Æ dt), where t = 1, 2, . . . , (p 1)/2. Then expression (36) has the form Lp ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðc1 þ i d 1 Þn þ ðc1 i d 1 Þn þ ðc2 þ i d 2 Þn þ ðc2 i d 2 Þn þ þ ðcðp1Þ=2 þ i d ðp1Þ=2 Þn þ ðcðp1Þ=2 i d ðp1Þ=2 Þn ; If we use the Moivre formula (22) and simplify as in (30), we have
ð38Þ
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0
1
c1 B C Lp ðnÞ ¼ ðx1 Þn þ xn2 cosðpnÞ þ 2ðc21 þ d 21 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þn=2 c21 þ d 21 0 1 0
1
cðp1Þ=2 c2 B C B C
cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ þ 2ðc2ðp1Þ=2 þ d 2ðp1Þ=2 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 cðp1Þ=2 þ d ðp1Þ=2 c2 þ d 2
ð39Þ
If we replace the discrete variable n with the continuous variable x in Eqs. (37) and (39), we have two formulas for the continuous functions for the Lucas p-numbers: For the even p 1 1 0 0 c1 c2 C C B B Lp ðxÞ ¼ ðx1 Þn þ 2ðc21 þ d 21 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2 2 2 2 c1 þ d 1 c2 þ d 2 0 1 cp=2 B C þ 2ðc2p=2 þ d 2p=2 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 2 cp=2 þ d p=2 For the odd p
0
ð40Þ 1
c1 C B Lp ðxÞ ¼ ðx1 Þx þ xx2 cosðpxÞ þ 2ðc21 þ d 21 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þx=2 2 2 c1 þ d 1 0 1 0
1
cðp1Þ=2 c2 B C B C
cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ þ ðc2ðp1Þ=2 þ d 2ðp1Þ=2 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 cðp1Þ=2 þ d ðp1Þ=2 c2 þ d 2
ð41Þ
5.2. The continuous functions for the Fibonacci p-numbers In [29], we see that the following analytical form represents the Fibonacci p-numbers: F p ðnÞ ¼ k 1 ðx1 Þn þ k 2 ðx2 Þn þ k 3 ðx3 Þn þ þ k p ðxp Þn þ k pþ1 ðxpþ1 Þn ;
ð42Þ
where x1, x2, x3, . . . , xp, xp+1 are the roots of Eq. (15) and k1, k2, k3, . . . , kp, kp + 1 are some constant coefficients that depend on the initial terms (12) of the Fibonacci p-series. These are solutions to the following system of algebraic equations: 8 F p ð0Þ ¼ k 1 þ k 2 þ k 3 þ k 4 þ þ k p þ k pþ1 ; > > > > > F p ð1Þ ¼ k 1 x1 þ k 2 x2 þ k 3 x3 þ þ k p xp þ k pþ1 xpþ1 ; > > > > > < F p ð2Þ ¼ k 1 ðx1 Þ2 þ k 2 ðx2 Þ2 þ k 3 ðx3 Þ2 þ þ k p ðxp Þ2 þ k pþ1 ðxpþ1 Þ2 ; ð43Þ F p ð3Þ ¼ k 1 ðx1 Þ3 þ k 2 ðx2 Þ3 þ k 3 ðx3 Þ3 þ þ k p ðxp Þ3 þ k pþ1 ðxpþ1 Þ3 ; > > > > > .. > > > . > > : F p ðpÞ ¼ k 1 ðx1 Þp þ k 2 ðx2 Þp þ k 3 ðx3 Þp þ þ k p ðxp Þp þ k pþ1 ðxpþ1 Þp ; where Fp(0) = 0 b Fp(1) = Fp(2) = Fp(3) = = Fp(p) = 1 are the initial terms of the Fibonacci p-series. Consider two possible cases. 1. If p is even, then the coefficient k1 is a real number and the rest of the p coefficients k2, k3, . . . , kp, kp+1 form the p/2 pairs of complex conjugate roots (at + i Æ bt) and (at i Æ bt), where t = 1, 2, 3, . . . , p/2}. Then expression (42) becomes F p ðnÞ ¼ k 1 ðx1 Þn þ ða1 i b1 Þðc1 þ i d 1 Þn þ ða1 þ i b1 Þðc1 i d 1 Þn þ ða2 i b2 Þðc2 þ i d 2 Þn þ ða2 þ i b2 Þðc2 i d 2 Þn þ þ ðap=2 i bp=2 Þðcp=2 þ i d p=2 Þn þ ðap=2 þ i bp=2 Þðcp=2 i d p=2 Þn .
ð44Þ
Consider the expression ða þ i bÞðc i dÞn þ ða i bÞðc þ i dÞn ¼ a½ðc i dÞn þ ðc þ i dÞn þ i b½ðc i dÞn ðc þ i dÞn .
ð45Þ
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1023
If we use the Moivre formula (22), we can write Eq. (45) as follows: 2aðc2 þ d 2 Þn=2 cosðnhÞ þ 2i bðc2 þ d 2 Þn=2 ðiÞ sinðnhÞ ¼ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ þ b sinðnhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ðc2 þ d 2 Þn=2 a2 þ b2 cosðnh þ cÞ; where h ¼ arccos
c ffi pffiffiffiffiffiffiffiffi c2 þd 2
; c ¼ arccos
a ffi pffiffiffiffiffiffiffiffi a2 þb2
ð46Þ
.
Using (46), we write expression (45) in the form: ffi ffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ b21 cosðnh1 þ c1 Þ þ 2 c22 þ d 22 a22 þ b22 cosðnh2 þ c2 Þ þ F p ðnÞ ¼ k 1 ðx1 Þn þ 2 c21 þ d 21
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2p=2 þ b2p=2 cosðnhp=2 þ cp=2 Þ. þ 2 c2p=2 þ d 2p=2
ð47Þ
2. If p is odd, then the coefficients k1 and k2 are real numbers and the rest of the p 1 coefficients form (p 1)/2 pairs of complex conjugate coefficients. We denote these pairs of complex conjugate coefficients by (at + i Æ bt) and (at i Æ bt), where t = 1, 2, 3, . . . , (p 1)/2. Then the expression (42) becomes: F p ðnÞ ¼ k 1 ðx1 Þn þ k 2 ðx2 Þn þ ða1 i b1 Þðc1 þ i d 1 Þn þ ða1 þ i b1 Þðc1 i d 1 Þn þ ða2 i b2 Þðc2 þ i d 2 Þn þ ða2 þ i b2 Þðc2 i d 2 Þn þ þ ½aðp1Þ=2 i bðp1Þ=2 ½cðp1Þ=2 þ i d ðp1Þ=2 n þ ½aðp1Þ=2 þ i bðp1Þ=2
½cðp1Þ=2 i d ðp1Þ=2 n .
ð48Þ
Using (46), we have the following expression for (48): ffi ffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ b21 cosðnh1 þ c1 Þ þ 2 c22 þ d 22 a22 þ b22 cosðnh2 þ c2 Þ þ F p ðnÞ ¼ k 1 ðx1 Þn þ ðx2 Þn cosðpnÞ þ 2 c21 þ d 21
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 c2ðp1Þ=2 þ d 2ðp1Þ=2 a2ðp1Þ=2 þ b2ðp1Þ=2 cosðnhðp1Þ=2 þ cðp1Þ=2 Þ. ð49Þ If we replace the discrete variable n with the continuous variable x in the expressions (47) and (49), we will get two formulas that give the continuous functions for the Fibonacci p-numbers: For the even p 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p=2 X
c a x=2 B C i i F p ðxÞ ¼ k 1 ðx1 Þx þ 2 a21 þ b21 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2i þ d 2i ð50Þ i¼1 c2i þ d 2i a2i þ b2i For the odd p x
F p ðxÞ ¼ k 1 ðx1 Þ þ ðx2 Þ cosðpxÞ þ 2
ðp1Þ=2 X i¼1
0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c a x=2 B C i i a21 þ b21 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2i þ d 2i 2 2 2 2 ci þ d i ai þ bi ð51Þ
6. Conclusion At first glance, the formulas obtained in the present article seem to be a mathematical abstraction that is very far from physical applications. However, let us recollect the primary mathematical meaning of the Fibonacci p-numbers (11) and (12), and the golden algebraic equation (15), which motivated our research. As we know, Stakhov [17] introduced the Fibonacci p-numbers within the research of the diagonal sums of the Pascal triangle and proved the following expression for the Fibonacci p-numbers: F p ðn þ 1Þ ¼ C 0n þ C 1np þ C 2n2p þ þ C mmþr .
ð52Þ
Note that Eq. (52) connects the Fibonacci p-numbers to binomial coefficients and expresses one more secret of the Pascal triangle. The fact that there exist so many different derivations of the Fibonacci p-series: the recurrence form of Eqs. (11) and (12), the analytical form of Eqs. (42), (49), and (51), and by the sum of binomial coefficients from Eq. (52) is astonishing and of fundamental interest.
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As we consider practical applications of Eq. (52), we note that for the case p = 0, the formula (52) is reduced to a widely known formula of from combinatorial analysis: 2n ¼ C 0n þ C 1n þ C 2n þ þ C nn .
ð53Þ
Note that Eq. (53) has a huge number of applications in different fields of modern science, particularly in coding theory. Let us review the origin of the golden algebraic equation (15). This equation resulted from research on the ratio of the adjacent Fibonacci p-numbers, that is, the ratio Fp(n)/Fp(n 1). If we consider the limit of this ratio as n ! 1, then we will come to a new class of irrational numbers sp (p = 1, 2, 3, . . .) that are the roots of the algebraic equation (15). They are called the Golden p-proportions because, for the case p = 1, the number s1 coincides with the classic Golden Proportion. It follows from this consideration that the golden algebraic equation (15) keeps one more ‘‘secret’’ of the Pascal triangle. If, from such positions, we consider the mathematical formulas for the continuous functions for the Fibonacci and Lucas p-numbers, which we express using the roots of the algebraic equation (15), then we may assert that these formulas are of fundamental interest in both the development of combinatorial analysis and the development of the Fibonacci number theory [14–16]. In addition, they express some deep regularities of the physical world; an idea which others began to develop in works [1–13].
Acknowledgements Authors most grateful to Dr. Pamela Ryan (Milligan College) for help in translating this article into the English.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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[24] Stakhov AP. Hyperbolic Fibonacci and Lucas functions: a new mathematics for the living nature. Vinnitsa: ITI; 2003. [25] Stakhov AP. The generalized golden sections and a new approach to the geometric definition of a number. Ukrainian Math J 2004;56(8):1143–50 [in Russian]. [26] Stakhov A, Rozin B. On a new class of hyperbolic function. Chaos, Solitons & Fractals 2005;23(2):379–89. [27] Stakhov AP. The generalized principle of the golden section and its applications in mathematics, science, and engineering. Chaos, Solitons & Fractals 2005;26(2):263–89. [28] Stakhov A, Rozin B. The Golden Shofar. Chaos, Solitons & Fractals 2005;26(3):677–84. [29] Stakhov A, Rozin B. Theory of Binet formulas for Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals 2005;27(5):1162–77. [30] Stakhov A, Rozin B. The ‘‘golden’’ algebraic equations. Chaos, Solitons & Fractals 2005;27(5):1415–21.
The continuous functions for the Fibonacci and Lucas p-numbers Alexey Stakhov *, Boris Rozin International Club of the Golden Section, 6 McCreary Trail, Bolton, ON, Canada L7E 2C8 Accepted 30 August 2005
Abstract The new continuous functions for the Fibonacci and Lucas p-numbers using Binet formulas are introduced. The article is of a fundamental interest for Fibonacci numbers theory and theoretical physics. Ó 2005 Elsevier Ltd. All rights reserved.
1. New results in Fibonacci number theory Modern science, particularly physics [1–13], widely applies the recurring series of the Fibonacci numbers F(n) f0; 1; 1; 2; 3; 5; 8; 13; 21; 34; . . .g
ð1Þ
and the recurring series of the Lucas numbers L(n) f2; 1; 3; 4; 7; 11; 18; 29; 47; . . .g
ð2Þ
which result from application of the following recurrence relations: F ðnÞ ¼ F ðn 1Þ þ F ðn 2Þ for n > 1; F ð0Þ ¼ 0; F ð1Þ ¼ 1; LðnÞ ¼ Lðn 1Þ þ Lðn 2Þ for n > 1; Lð0Þ ¼ 2; Lð1Þ ¼ 1.
ð3Þ ð4Þ ð5Þ ð6Þ
The ratio pffiffi of the adjacent numbers in the Fibonacci series (1) and Lucas series (2) tends toward at the irrational number s ¼ 1þ2 5 called the Golden Proportion (Golden Mean) [14–16]. In the 19th century the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers [14–16]:
*
Corresponding author. E-mail addresses: [email protected] (A. Stakhov), [email protected] (B. Rozin). URLs: http://www.goldenmuseum.com (A. Stakhov), http://www.goldensection.net (B. Rozin).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.158
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sn ð1Þn sn pffiffiffi ; 5 LðnÞ ¼ sn þ ð1Þn sn ; F ðnÞ ¼
ð7Þ ð8Þ
where s is the Golden Proportion and n = 0, ±1, ±2, ±3 . . . In recent years, researchers developed further the theory of the Golden Section and the Fibonacci numbers [17–30]. Hyperbolic Fibonacci and Lucas functions [21,24,26], which are extensions of Binet formulas for a continuous domain, have a strategic importance for the development of both mathematics and theoretical physics. In a previous article [28], the authors presented a new function of the second power called the Golden Shofar. The graph of this function reminds one of a horn, which is used to blow in the Yom Kippur (The Judgment Day). The sinusoidal Fibonacci function [28] is a further development of a continuous approach to the Fibonacci numbers theory begun in a series of papers [21,24,26]. The definition of the function is F ðxÞ ¼
sx cosðpxÞsx pffiffiffi 5
ð9Þ
Fig. 1. The sinusoidal Fibonacci function. Table 1 The extended Fibonacci and Lucas numbers N F(n) F(n) L(n) L(n)
0 0 0 2 2
1 1 1 1 1
2 1 1 3 3
3 2 2 4 4
4 3 3 7 7
5 5 5 11 11
6 8 8 18 18
7 13 13 29 29
8 21 21 47 47
9 34 34 76 76
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The graph of the sinusoidal Fibonacci function given by Eq. (9) is shown in Fig. 1. Note that for the discrete values of the continuous variable x = n = 0, ±1, ±2, ±3, . . . the function (9) reduces to expression (7). That is, in the discrete points x = n = 0, ±1, ±2, ±3 . . . the values of the sinusoidal Fibonacci function (9) coincide with the extended Fibonacci series (Table 1) and, clearly, the extended Fibonacci numbers are inscribed into the graph of the function in Fig. 1. Similarly, we can introduce a sinusoidal Lucas function: LðxÞ ¼ sx þ cosðpxÞsx ;
ð10Þ
that reduces to expression (8) for discrete values of x = n = 0, ±1, ±2, ±3, . . .. It is again clear that the extended Lucas numbers (Table 1) are inscribed to the graph of the function (10). The introduction of generalized Fibonacci and Lucas numbers or Fibonacci and Lucas p-numbers [17,29] is the next new result within the Fibonacci numbers theory. It is proved in [17] that, for a given integer p, p = 1, 2, 3, . . ., the Fibonacci p-numbers Fp(n) are given by the following recurrence formula: F p ðnÞ ¼ F p ðn 1Þ þ F p ðn p 1Þ for n > p
ð11Þ
with the initial conditions: F p ð0Þ ¼ 0;
F p ð1Þ ¼ F p ð2Þ ¼ ¼ F p ðpÞ ¼ 1;
ð12Þ
where n = 0, ±1, ±2, ±3, . . . In article [28], the authors introduced a new class of recurrent numerical sequences, called the Lucas p-numbers, Lp(n), given by the recurrence relation: Lp ðnÞ ¼ Lp ðn 1Þ þ Lp ðn p 1Þ for n > p
ð13Þ
for a given integer p, p = 1, 2, 3, . . ., and with the initial conditions: Lp ð0Þ ¼ p þ 1;
Lp ð1Þ ¼ Lp ð2Þ ¼ ¼ Lp ðpÞ ¼ 1;
ð14Þ
where n = 0, ±1, ±2, ±3, . . . Note that for p = 1 the classic Fibonacci numbers (1) coincide with the Fibonacci p-numbers and the classic Lucas numbers (2) coincide with the Lucas p-numbers. Hence, the introduction of the Fibonacci and Lucas p-numbers considerably develops this area of Fibonaccis research [14–16]. The original research presented in the article Theory of Binet formulas for the Fibonacci and Lucas p-numbers [29] is a new mathematical result for the Fibonacci numbers theory. In essence, for the first time since Binet derived his formulas, we have derived analytical formulas that allow us to express numerical sequences given by Eqs. (11), (12) and (13), (14) using the roots of the golden algebraic equation: xpþ1 ¼ xp þ 1.
ð15Þ
For the case p = 1, the new Binet formulas correspond to the Binet formulas (7), (8) for the classic Fibonacci and Lucas numbers. The purpose of the present article is to develop continuous functions within the Fibonacci number theory which lead to the hyperbolic Fibonacci and Lucas functions [21,24,26], to the function of the ‘‘Golden Shofar’’ [28], and the sinusoidal Fibonacci and Lucas functions (9), (10). The main goal of the article is to extend the results (9) and (10) for the area of the Fibonacci and Lucas p-numbers. That is, to derive formulas for the continuous functions for the Fibonacci and Lucas p-numbers.
2. The continuous functions for the Fibonacci and Lucas 2-numbers 2.1. The continuous function for the Lucas 2-numbers In [29] we defined the Lucas 2-numbers as follows: L2 ðnÞ ¼ L2 ðn 1Þ þ L2 ðn 3Þ; L2 ð0Þ ¼ 3; L2 ð1Þ ¼ L2 ð2Þ ¼ 1.
ð16Þ ð17Þ
For the case p = 2, the golden algebraic equation takes the following form: x3 x2 1 ¼ 0.
ð18Þ
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This equation has three roots: 2 h þ 2h þ 4 x1 ¼ ; 6h
pffiffiffi ! h 1 1 3 h 2 ; x2 ¼ þ i 12 3h 3 2 6 3h pffiffiffi ! h 1 1 3 h 2 x2 ¼ þ þ i ; 2 6 3h 12 3h 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffi 3 where h ¼ ð116 þ 12 93Þ. In [29], we derive the following Binet formula for the Lucas 2-numbers: L2 ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðx3 Þn ; where n = 0, ±1, ±2, ±3, . . .; x1, x2, x3 are the roots of the algebraic equation (18) given by (19). In the interest of clarity and simplicity, we use the following designations: pffiffiffi h 1 1 3 h 2 . c¼ þ ; d¼ 12 3h 3 2 6 3h
ð19Þ
ð20Þ
ð21Þ
We will use the Moivre formula: ðc i dÞn ¼ ðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ ; where h ¼ arccos
c ffi pffiffiffiffiffiffiffiffi c2 þd 2
ð22Þ
.
If we substitute (22) into (19) and then (19) into (20), we will get 2 n h þ 2h þ 4 L2 ðnÞ ¼ þ ðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ þ ðc2 þ d 2 Þn=2 ½cosðnhÞ þ i sinðnhÞ 6h 2 n h þ 2h þ 4 ¼ þ 2ðc2 þ d 2 Þn=2 cosðnhÞ 6h " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n 2 n h þ 2h þ 4 ðh 2Þ h2 þ 2h þ 4 h2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p n ; þ cos arccos ¼ 6h 6h 2 h2 þ 2h þ 4
ð23Þ
where n = 0, ±1, ±2, ±3, . . . is a discrete variable. If we replace the discrete variable n with the continuous variable x in the expression (23), we will get the following continuous function called the continuous function for the Lucas 2-numbers: " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x 2 x h þ 2h þ 4 ðh 2Þ h2 þ 2h þ 4 2h L2 ðxÞ ¼ ð24Þ þ cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x . 6h 6h 2 h2 þ 2h þ 4 The graph of the function (24) is shown in Fig. 2. 2.2. The continuous function for the Fibonacci 2-numbers In [17] we find the following definition of the Fibonacci 2-numbers: F 2 ðnÞ ¼ F 2 ðn 1Þ þ F 2 ðn 3Þ for n > 3; F 2 ð0Þ ¼ 0; F 2 ð1Þ ¼ F 2 ð2Þ ¼ 1. The Binet formula for the Fibonacci 2-numbers has the following form [29]: !n pffiffiffi pffiffiffi 2 n 2hðh þ 2Þ h2 þ 2h þ 4 ððh þ 2Þ þ i 3ðh 2ÞÞh h2 4h þ 4 3ðh 4Þ þ i F 2 ðnÞ ¼ 3 6h 12h 12h ðh þ 8Þ ðh3 þ 8Þ !n pffiffiffi
p ffiffi ffi ðh þ 2Þ i 3ðh 2Þ h h2 4h þ 4 3ðh2 4Þ þi þ ; 3 12h 12h ðh þ 8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ffi 3 where h ¼ ð116 þ 12 93Þ.
ð25Þ ð26Þ
ð27Þ
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Fig. 2. The continuous function for the Lucas 2-numbers.
In addition to (21), we introduce the following designations: pffiffiffi ðh þ 2Þh 3ðh 2Þh ; b ¼ . a¼ ðh3 þ 8Þ ðh3 þ 8Þ Then expression (27) becomes n 2hðh þ 2Þ h2 þ 2h þ 4 þ ða þ ibÞðc idÞn þ ða ibÞðc þ idÞn . F 2 ðnÞ ¼ 3 6h ðh þ 8Þ
ð28Þ
ð29Þ
If we use the Moivre formula (22), then we have following expression for (29): n 2hðh þ 2Þ h2 þ 2h þ 4 þ ða þ i bÞðc2 þ d 2 Þn=2 ½cosðnhÞ i sinðnhÞ F 2 ðnÞ ¼ 3 6h ðh þ 8Þ þ ða i bÞðc2 þ d 2 Þn=2 ½cosðnhÞ þ i sinðnhÞ n 2hðh þ 2Þ h2 þ 2h þ 4 ¼ 3 þ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ bðiÞ2 sinðnhÞ 6h ðh þ 8Þ n 2hðh þ 2Þ h2 þ 2h þ 4 þ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ þ b sinðnhÞ ¼ 3 6h ðh þ 8Þ n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hðh þ 2Þ h2 þ 2h þ 4 þ 2 a2 þ b2 ðc2 þ d 2 Þn=2 cosðnh gÞ; ¼ 3 6h ðh þ 8Þ
ð30Þ
a ffi where g ¼ arccos pffiffiffiffiffiffiffiffi . a2 þb2 Therefore,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n n 2hðh þ 2Þ h2 þ 2h þ 4 4h h2 2h þ 4 ðh 2Þ h2 þ 2h þ 4 F 2 ðnÞ ¼ 3 þ 6h 6h ðh þ 8Þ h3 þ 8 " ! !# 2h hþ2
cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 h2 þ 2h þ 4 2 h2 2h þ 4
ð31Þ
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Fig. 3. The continuous function for the Fibonacci 2-numbers.
If we replace the discrete variable n with the continuous variable x in Eq. (31), then we get the following continuous function called the continuous function for the Fibonacci 2-numbers: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x x 2hðh þ 2Þ h2 þ 2h þ 4 4h h2 2h þ 4 ðh 2Þ h2 þ 2h þ 4 F 2 ðxÞ ¼ 3 þ 6h 6h ðh þ 8Þ h3 þ 8 " ! !# 2h hþ2 ð32Þ
cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 h2 þ 2h þ 4 2 h2 2h þ 4 Fig. 3 is the graph of function (32). The continuous functions for the Fibonacci and Lucas 2-numbers have recurrent properties that are similar to the Fibonacci and Lucas 2-numbers. Recurrent properties for the Fibonacci and Lucas 2-numbers
Recurrent properties for the continuous functions for the Fibonacci and Lucas 2-numbers
F2(n) = F2(n 1) + F2(n 3) L2(n) = L2(n 1) + L2(n 3)
F2(x) = F2(x 1) + F2(x 3) L2(x) = L2(x 1) + L2(x 3)
3. The continuous functions for the Fibonacci and Lucas 3-numbers If we use the same reasoning for the Fibonacci and Lucas 3-numbers [17,29], we derive the following continuous functions for these numerical series: L3 ðxÞ ¼ 1:38x þ ð0:819Þx cosðpxÞ þ ð0; 9399Þx cosð1:335xÞ; F 3 ðxÞ ¼ 0:3969 1:38x 0:1592 ð0:819Þx cosðpxÞ þ 0:2365 ð0:9399Þx cosð1:335x 2:097Þ.
ð33Þ ð34Þ
Their graphs are shown in Figs. 4 and 5. The continuous functions for the Fibonacci and Lucas 3-numbers have recurrent properties that are similar to the Fibonacci and Lucas 3-numbers.
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Recurrent properties for the Fibonacci and Lucas 3-numbers
Recurrent properties for the continuous functions for the Fibonacci and Lucas 3-numbers
F3(n) = F3(n 1) + F3(n 4) L3(n) = L3(n 1) + L3(n 4)
F3(x) = F3(x 1) + F3(x 4) L3(x) = L3(x 1) + L3(x 4)
Fig. 4. The continuous function for the Lucas 3-numbers.
Fig. 5. The continuous function for the Fibonacci 3-numbers.
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1021
4. The continuous functions for the Fibonacci and Lucas 4-numbers The continuous functions for the Fibonacci and Lucas 4-numbers [17,29] have the following form: " ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!x x px h4 12h2 þ 144 h 2 h2 þ 12 þ 2 cos cos arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ; L4 ðxÞ ¼ þ þ2 6 h 3 6h 2 h4 12h2 þ 144 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h 2 px þ g1 þ 2 a22 þ b22 ðc22 þ d 22 Þx=2 cosðh2 x g2 Þ; þ 2 a21 þ b21 cos F 4 ðxÞ ¼ k 1 6 h 3 where h¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 3 ð108 þ 12 69Þ;
k 1 ¼ 0:38095;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 ¼ arccos a1 a21 þ b21 ;
h2 þ 12 h2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 h4 12h2 þ 144
a2 ¼ 0:1191;
b2 ¼ 0:04577;
a1 ¼ 0:07133;
b1 ¼ 0:2063;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 ¼ arccos a2 a22 þ b22 .
5. The continuous functions for the Fibonacci and Lucas p-numbers (a general case) 5.1. The continuous functions for the Lucas p-numbers The Lucas p-numbers (13) and (14) were introduced in [29]. There we find the proof that the following analytical form, which is the Binet formula for the Lucas p-numbers, represents the Lucas p-numbers: Lp ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðx3 Þn þ þ ðxp Þn þ ðxpþ1 Þn ;
ð35Þ
where x1, x2, x3, . . . , xp, xp+1 are the roots of Eq. (15). If p is even, then Eq. (15) has the only the real root x1, which is equal to the Golden p-proportion sp and p/2 pairs of complex conjugate roots. Let us denote the pair of complex conjugate roots by (ct + i Æ dt) and (ct i Æ dt), where t = 1, 2, . . . , p/2. Then Eq. (35) becomes Lp ðnÞ ¼ ðx1 Þn þ ðc1 þ i d 1 Þn þ ðc1 i d 1 Þn þ ðc2 þ i d 2 Þn þ ðc2 i d 2 Þn þ þ ðcp=2 þ i d p=2 Þn þ ðcp=2 i d p=2 Þn . If we use the Moivre formula (22) and simplify, as in (30), we see that 0 1
0
ð36Þ
1
c1 c2 B C B C Lp ðnÞ ¼ ðx1 Þn þ 2ðc21 þ d 21 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2 2 c21 þ d 1 c22 þ d 2 0 1 cp=2 B C þ 2ðc2p=2 þ d 2p=2 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2p=2 þ d 2p=2
ð37Þ
If p is odd, then Eq. (15) has two real roots, x1 and x2, which have opposite signs. In that case, the positive root x1 coincides with the Golden p-proportion sp; the rest of the p 1 roots are the (p 1)/2 pairs of complex conjugate roots. We denote the pair of complex conjugate roots by (ct + i Æ dt) and (ct i Æ dt), where t = 1, 2, . . . , (p 1)/2. Then expression (36) has the form Lp ðnÞ ¼ ðx1 Þn þ ðx2 Þn þ ðc1 þ i d 1 Þn þ ðc1 i d 1 Þn þ ðc2 þ i d 2 Þn þ ðc2 i d 2 Þn þ þ ðcðp1Þ=2 þ i d ðp1Þ=2 Þn þ ðcðp1Þ=2 i d ðp1Þ=2 Þn ; If we use the Moivre formula (22) and simplify as in (30), we have
ð38Þ
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0
1
c1 B C Lp ðnÞ ¼ ðx1 Þn þ xn2 cosðpnÞ þ 2ðc21 þ d 21 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þn=2 c21 þ d 21 0 1 0
1
cðp1Þ=2 c2 B C B C
cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ þ 2ðc2ðp1Þ=2 þ d 2ðp1Þ=2 Þn=2 cos @n arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 cðp1Þ=2 þ d ðp1Þ=2 c2 þ d 2
ð39Þ
If we replace the discrete variable n with the continuous variable x in Eqs. (37) and (39), we have two formulas for the continuous functions for the Lucas p-numbers: For the even p 1 1 0 0 c1 c2 C C B B Lp ðxÞ ¼ ðx1 Þn þ 2ðc21 þ d 21 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2 2 2 2 c1 þ d 1 c2 þ d 2 0 1 cp=2 B C þ 2ðc2p=2 þ d 2p=2 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 2 cp=2 þ d p=2 For the odd p
0
ð40Þ 1
c1 C B Lp ðxÞ ¼ ðx1 Þx þ xx2 cosðpxÞ þ 2ðc21 þ d 21 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ 2ðc22 þ d 22 Þx=2 2 2 c1 þ d 1 0 1 0
1
cðp1Þ=2 c2 B C B C
cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ þ ðc2ðp1Þ=2 þ d 2ðp1Þ=2 Þx=2 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 cðp1Þ=2 þ d ðp1Þ=2 c2 þ d 2
ð41Þ
5.2. The continuous functions for the Fibonacci p-numbers In [29], we see that the following analytical form represents the Fibonacci p-numbers: F p ðnÞ ¼ k 1 ðx1 Þn þ k 2 ðx2 Þn þ k 3 ðx3 Þn þ þ k p ðxp Þn þ k pþ1 ðxpþ1 Þn ;
ð42Þ
where x1, x2, x3, . . . , xp, xp+1 are the roots of Eq. (15) and k1, k2, k3, . . . , kp, kp + 1 are some constant coefficients that depend on the initial terms (12) of the Fibonacci p-series. These are solutions to the following system of algebraic equations: 8 F p ð0Þ ¼ k 1 þ k 2 þ k 3 þ k 4 þ þ k p þ k pþ1 ; > > > > > F p ð1Þ ¼ k 1 x1 þ k 2 x2 þ k 3 x3 þ þ k p xp þ k pþ1 xpþ1 ; > > > > > < F p ð2Þ ¼ k 1 ðx1 Þ2 þ k 2 ðx2 Þ2 þ k 3 ðx3 Þ2 þ þ k p ðxp Þ2 þ k pþ1 ðxpþ1 Þ2 ; ð43Þ F p ð3Þ ¼ k 1 ðx1 Þ3 þ k 2 ðx2 Þ3 þ k 3 ðx3 Þ3 þ þ k p ðxp Þ3 þ k pþ1 ðxpþ1 Þ3 ; > > > > > .. > > > . > > : F p ðpÞ ¼ k 1 ðx1 Þp þ k 2 ðx2 Þp þ k 3 ðx3 Þp þ þ k p ðxp Þp þ k pþ1 ðxpþ1 Þp ; where Fp(0) = 0 b Fp(1) = Fp(2) = Fp(3) = = Fp(p) = 1 are the initial terms of the Fibonacci p-series. Consider two possible cases. 1. If p is even, then the coefficient k1 is a real number and the rest of the p coefficients k2, k3, . . . , kp, kp+1 form the p/2 pairs of complex conjugate roots (at + i Æ bt) and (at i Æ bt), where t = 1, 2, 3, . . . , p/2}. Then expression (42) becomes F p ðnÞ ¼ k 1 ðx1 Þn þ ða1 i b1 Þðc1 þ i d 1 Þn þ ða1 þ i b1 Þðc1 i d 1 Þn þ ða2 i b2 Þðc2 þ i d 2 Þn þ ða2 þ i b2 Þðc2 i d 2 Þn þ þ ðap=2 i bp=2 Þðcp=2 þ i d p=2 Þn þ ðap=2 þ i bp=2 Þðcp=2 i d p=2 Þn .
ð44Þ
Consider the expression ða þ i bÞðc i dÞn þ ða i bÞðc þ i dÞn ¼ a½ðc i dÞn þ ðc þ i dÞn þ i b½ðc i dÞn ðc þ i dÞn .
ð45Þ
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1023
If we use the Moivre formula (22), we can write Eq. (45) as follows: 2aðc2 þ d 2 Þn=2 cosðnhÞ þ 2i bðc2 þ d 2 Þn=2 ðiÞ sinðnhÞ ¼ 2ðc2 þ d 2 Þn=2 ½a cosðnhÞ þ b sinðnhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ðc2 þ d 2 Þn=2 a2 þ b2 cosðnh þ cÞ; where h ¼ arccos
c ffi pffiffiffiffiffiffiffiffi c2 þd 2
; c ¼ arccos
a ffi pffiffiffiffiffiffiffiffi a2 þb2
ð46Þ
.
Using (46), we write expression (45) in the form: ffi ffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ b21 cosðnh1 þ c1 Þ þ 2 c22 þ d 22 a22 þ b22 cosðnh2 þ c2 Þ þ F p ðnÞ ¼ k 1 ðx1 Þn þ 2 c21 þ d 21
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2p=2 þ b2p=2 cosðnhp=2 þ cp=2 Þ. þ 2 c2p=2 þ d 2p=2
ð47Þ
2. If p is odd, then the coefficients k1 and k2 are real numbers and the rest of the p 1 coefficients form (p 1)/2 pairs of complex conjugate coefficients. We denote these pairs of complex conjugate coefficients by (at + i Æ bt) and (at i Æ bt), where t = 1, 2, 3, . . . , (p 1)/2. Then the expression (42) becomes: F p ðnÞ ¼ k 1 ðx1 Þn þ k 2 ðx2 Þn þ ða1 i b1 Þðc1 þ i d 1 Þn þ ða1 þ i b1 Þðc1 i d 1 Þn þ ða2 i b2 Þðc2 þ i d 2 Þn þ ða2 þ i b2 Þðc2 i d 2 Þn þ þ ½aðp1Þ=2 i bðp1Þ=2 ½cðp1Þ=2 þ i d ðp1Þ=2 n þ ½aðp1Þ=2 þ i bðp1Þ=2
½cðp1Þ=2 i d ðp1Þ=2 n .
ð48Þ
Using (46), we have the following expression for (48): ffi ffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ b21 cosðnh1 þ c1 Þ þ 2 c22 þ d 22 a22 þ b22 cosðnh2 þ c2 Þ þ F p ðnÞ ¼ k 1 ðx1 Þn þ ðx2 Þn cosðpnÞ þ 2 c21 þ d 21
n=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 c2ðp1Þ=2 þ d 2ðp1Þ=2 a2ðp1Þ=2 þ b2ðp1Þ=2 cosðnhðp1Þ=2 þ cðp1Þ=2 Þ. ð49Þ If we replace the discrete variable n with the continuous variable x in the expressions (47) and (49), we will get two formulas that give the continuous functions for the Fibonacci p-numbers: For the even p 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p=2 X
c a x=2 B C i i F p ðxÞ ¼ k 1 ðx1 Þx þ 2 a21 þ b21 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2i þ d 2i ð50Þ i¼1 c2i þ d 2i a2i þ b2i For the odd p x
F p ðxÞ ¼ k 1 ðx1 Þ þ ðx2 Þ cosðpxÞ þ 2
ðp1Þ=2 X i¼1
0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c a x=2 B C i i a21 þ b21 cos @x arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ arccos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. c2i þ d 2i 2 2 2 2 ci þ d i ai þ bi ð51Þ
6. Conclusion At first glance, the formulas obtained in the present article seem to be a mathematical abstraction that is very far from physical applications. However, let us recollect the primary mathematical meaning of the Fibonacci p-numbers (11) and (12), and the golden algebraic equation (15), which motivated our research. As we know, Stakhov [17] introduced the Fibonacci p-numbers within the research of the diagonal sums of the Pascal triangle and proved the following expression for the Fibonacci p-numbers: F p ðn þ 1Þ ¼ C 0n þ C 1np þ C 2n2p þ þ C mmþr .
ð52Þ
Note that Eq. (52) connects the Fibonacci p-numbers to binomial coefficients and expresses one more secret of the Pascal triangle. The fact that there exist so many different derivations of the Fibonacci p-series: the recurrence form of Eqs. (11) and (12), the analytical form of Eqs. (42), (49), and (51), and by the sum of binomial coefficients from Eq. (52) is astonishing and of fundamental interest.
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As we consider practical applications of Eq. (52), we note that for the case p = 0, the formula (52) is reduced to a widely known formula of from combinatorial analysis: 2n ¼ C 0n þ C 1n þ C 2n þ þ C nn .
ð53Þ
Note that Eq. (53) has a huge number of applications in different fields of modern science, particularly in coding theory. Let us review the origin of the golden algebraic equation (15). This equation resulted from research on the ratio of the adjacent Fibonacci p-numbers, that is, the ratio Fp(n)/Fp(n 1). If we consider the limit of this ratio as n ! 1, then we will come to a new class of irrational numbers sp (p = 1, 2, 3, . . .) that are the roots of the algebraic equation (15). They are called the Golden p-proportions because, for the case p = 1, the number s1 coincides with the classic Golden Proportion. It follows from this consideration that the golden algebraic equation (15) keeps one more ‘‘secret’’ of the Pascal triangle. If, from such positions, we consider the mathematical formulas for the continuous functions for the Fibonacci and Lucas p-numbers, which we express using the roots of the algebraic equation (15), then we may assert that these formulas are of fundamental interest in both the development of combinatorial analysis and the development of the Fibonacci number theory [14–16]. In addition, they express some deep regularities of the physical world; an idea which others began to develop in works [1–13].
Acknowledgements Authors most grateful to Dr. Pamela Ryan (Milligan College) for help in translating this article into the English.
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