Download Fibonacci Numbers and Chebyshev Polynomials Takahiro Yamamoto December 2, 2015

Fibonacci Numbers and Chebyshev Polynomials
Takahiro Yamamoto
Department of Physics and Astronomy, University of Utah, USA
December 2, 2015
Abstract
The relation between the Fibonacci Sequence and the Golden Ratio
is · · · seemingly unrelated topics.
1
1.1
Fibonacci Sequence and Golden Ratio
Recurrence Relation
The Fibonacci numbers: 0, 1, 1, 2, 3, 5, · · ·, are the sequence of numbers defined by the linear recurrence equation:
fn = fn−1 + fn−2 .
(1)
To find fn , we first rewrite Eq. 1 in the matrix form:
fn
fn−1
1 1
n−1 f1
=M
= ··· = M
.
, where M =
fn−1
fn−2
f0
1 0
Noting that f0 = 0 and f1 = 1, we obtain fn = (M n )1,2 = (M n−1 )1,1 .
The characteristic equation of M is given by
λ2 − λ − 1 = 0,
and by solving Eq. 2, you obtain the eigenvalues
√
1± 5
,
λ± =
2
(2)
(3)
and the closed form of fn is therefore given by
fn =
λn+ − λn−
√
5
1
(4)
The Golden Spiral
10
r(θ) = exp
0
2 ln(ϕ)
π θ
−10
−20
−30
−10
0
10
20
30
40
Figure 1: The left panel is the image of the Whirlpool Galaxy (M51), as
a spectacular examples of the appearance of the gold ratio in nature. The
right panel is the plot of Eq. 7 in polar coordinates for θ ∈ [0, 4π].
1.2
The “Golden Ratio”
√
λ+ = (1 + 5)/2 = 1.6180 · · · is widely acknowledged as the Golden Ratio,
often denoted as ϕ. Since ϕ is the roots of Eq. 2, it can be expressed in a
nested radical representation:
p
ϕ = 1 + ϕ = ···
(5)
and can also be expressed in the form of continued fraction:
1
ϕ = 1 + = ···
ϕ
(6)
The golden ratio appears frequently in nature [1] (e.g. in the structure
of crystals) , and the special case of logarithmic spirals
2 ln(ϕ)
r(θ) = exp
θ ,
(7)
π
is often called the Golden Spiral, which can be observed in the arms of the
spiral galaxies as illustrated in Fig. 1, a shape nautilus shell and so forth.
2
2.1
Chebyshev Polynomial
Chebyshev Polynomial of the first kind
The Chebyshev polynomials of the first kind · · · .
The first few polynomials are shown in Fig. 2 for n = 1, · · · , 5 and x ∈
2
Chebyshev Polynomial of the First Kind
1
T1 (x)
T2 (x)
T3 (x)
T4 (x)
T5 (x)
1
2
0
− 12
−1
−1
− 12
1
2
0
1
x
Figure 2: The first few Chebyshev polynomials of the first kind Tn (x) for
n = 1, · · · , 5 and x ∈ [−1, 1].
[−1, 1]. · · · . They appear in many brunches of mathematics, but most commonly known to be connected with trigonometric multiple-angle formulas
cos(nθ) = Tn (cos(θ)).
2.2
(8)
Chebyshev Polynomial of the second kind
The Chebyshev polynomials of the second kind1 , Un (x), defined as
∞
X
1
=
Un (x)tn ,
2
1 − 2tx + t
(9)
n=0
are a set of orthogonal polynomials:
Z 1
p
π
dxUm (x)Un (x) 1 − x2 = δm,n
2
−1
1
For more details, see Ref.[2]
3
(10)
They satisfy the same recurrence relations with Tn (x):
Un+1 (x) = 2xUn (x) − Un−1 (x).
(11)
with a slightly different initial conditions.
3
The Connection between the Fibonacci Sequence
and the Chebyshev polynomials of the second
kind
The Fibonacci numbers can be expressed in terms of the Chebyshev polynomial of the second kind by
fn+1 = i−n Un (i/2).
(12)
This rather bizarre relation has an elegant proof led by combinatorial models[3].
· · · , which provides the proof of Eq. 12.
References
[1] M. Livio, The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number, Broadway Books (2003)
[2] T. J. Rivlin, Chebyshev Polynomials: From Approximation Theory to
Algebra and Number Theory, Wiley-Interscience (1990)
[3] A. T. Benjamin and D. Walton, Counting on Chebyshev Polynomials,
Math. Mag. 82, No. 2, 117-126 (2009)
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