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 quite intriguing, given that the sequence of integer is somehow closely
related to the irrational number. More surprisingly, Chebyshev Polynomial of the second kind is also deeply connected to the Fibonacci
Sequence. In this article, we investigate the relation between those
seemingly unrelated topics.
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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
1
(2)
(3)
and the closed form of fn is therefore given by
fn =
1.2
λn+ − λn−
√
5
(4)
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:
v
s
u
r
u
q
p
√
t
ϕ = 1 + ϕ = ··· = 1 + 1 + 1 + 1 + 1 + ···
(5)
and can also be expressed in the form of continued fraction:
ϕ=1+
1
= ··· = 1 +
ϕ
1+
1
1
1+
1+
(6)
1
1
1+
1
1+
..
.
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
Chebyshev Polynomial
2.1
Chebyshev Polynomial of the first kind
The Chebyshev polynomials of the first kind1 , Tn (x), defined as
∞
X
1 − tx
=
Tn (x)tn ,
1 − 2tx + t2
n=0
1
For more details, see Ref.[2]
2
(8)
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].
are a set of orthogonal polynomials:
(
Z 1
1
πδm,n
Tm (x)Tn (x)
dx √
= 2
2
π
1−x
−1
for m 6= 0, n 6= 0
for m = n = 0
(9)
The first few polynomials are shown in Fig. 2 for n = 1, · · · , 5 and x ∈
[−1, 1]. They satisfy the recurrence relations:
Tn+1 (x) = 2xTn (x) − Tn−1 (x).
(10)
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
(11)
Chebyshev Polynomial of the second kind
The Chebyshev polynomials of the second kind2 , Un (x), defined as
∞
X
1
=
Un (x)tn ,
1 − 2tx + t2
n=0
2
For more details, see Ref.[2]
3
(12)
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].
are a set of orthogonal polynomials:
Z 1
p
π
dxUm (x)Un (x) 1 − x2 = δm,n
2
−1
(13)
They satisfy the same recurrence relations with Tn (x):
Un+1 (x) = 2xUn (x) − Un−1 (x).
(14)
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).
4
(15)
Figure 3: The number of square-domino fillings of length four: with no
weight, this corresponds to f5 = 4 C0 + 3 C1 + 2 C2 = 5, whilst with the
weight of 2x and −1 for square and domino respectably, this corresponds to
U4 (x) = 4 C0 (2x)4 + 3 C1 (−1)(2x)2 + 2 C2 (−1)2 = 16x4 − 12x2 + 1.
This rather bizarre relation has an elegant proof led by combinatorial models[3].
In the combinatorial model, the Fibonacci number fn+1 counts the ways to
fill a 1 × n stripe using 1 × 1 square and 1 × 2 dominos. As it turns out,
Chebyshev polynomials counts the same objects as the Fibonacci numbers,
with an additional weight to each square and domino. More specifically,
each square tile and domino are assigned a weight of 2x and −1 respectably.
Fig. 3 illustrates the counting of tilling of the length four stripe, which, with
no weighting, corresponds to f5 and which, with the weighting mentioned
above, corresponds to U4 (x). If we take x = i/2, we see that every tiling of
length n with k dominos has weight (−1)k in−2k = in , which turns out to be
independent of k. Since fn is the tiling of of length n, Un (i/2) = in fn+1 ,
which provides the proof of Eq. 15.
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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