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Fibonacci polynomial and the Pascal
Triangle
In mathematics polynomial functions, or
polynomials, are an important class of simple and smooth functions.
Simple means they are constructed using only multiplication and
addition. Smooth means they are infinitely differentiable, i.e.,
they have derivatives of all finite orders.
The Fibonacci polynomials are defined by the
recurrence relation
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(1) |
with  and  . They are also given by the explicit sum formula
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(2) |
where  is the floor function and  is a
binomial coefficient.
The Fibonacci numbers are recovered by evaluating
the polynomials at x = 1.
The first few Fibonacci polynomials are:
| F1(x) |
= |
1x0 |
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| F2(x) |
= |
1x1 |
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| F3(x) |
= |
1x2 |
+ |
1x0 |
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| F4(x) |
= |
1x3 |
+ |
2x1 |
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| F5(x) |
= |
1x4 |
+ |
3x2 |
+ |
1x0 |
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| F6(x) |
= |
1x5 |
+ |
4x3 |
+ |
3x1 |
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| F7(x) |
= |
1x6 |
+ |
5x4 |
+ |
6x2 |
+ |
1x0 |
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| F8(x) |
= |
1x7 |
+ |
6x5 |
+ |
10x3 |
+ |
4x1 |
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| F9(x) |
= |
1x8 |
+ |
7x6 |
+ |
15x4 |
+ |
10x2 |
+ |
1x0 |
| F10(x) |
= |
1x9 |
+ |
8x7 |
+ |
21x5 |
+ |
20x2 |
+ |
5x1 |
| ..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
It is obvious that Pascal`s Triangle structure is
built in these relations, which certainly indicates the existing
connection between the numbers of Pascal`s Triangle and Fibonacci
polynomials .
The Lucas polynomials are defined by the recurrence
relation
Ln+1(x)=x*Ln(x) +
Ln-1(x)
with Ln(0)=2 and
Ln(1)=x.
The Lucas numbers are recovered by evaluating the
polynomials at x = 1.The first few Lucas polynomials are:
| L1(x) |
= |
1x1 |
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| L2(x) |
= |
1x2 |
+ |
2x0 |
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| L3(x) |
= |
1x3 |
+ |
3x1 |
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| L4(x) |
= |
1x4 |
+ |
4x2 |
+ |
2x0 |
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| L5(x) |
= |
1x5 |
+ |
5x3 |
+ |
5x1 |
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| L6(x) |
= |
1x6 |
+ |
6x4 |
+ |
9x2 |
+ |
2x0 |
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| L7(x) |
= |
1x7 |
+ |
7x5 |
+ |
14x3 |
+ |
7x1 |
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| L8(x) |
= |
1x8 |
+ |
8x6 |
+ |
20x4 |
+ |
16x2 |
+ |
2x0 |
| L9(x) |
= |
1x9 |
+ |
9x7 |
+ |
27x5 |
+ |
30x3 |
+ |
9x1 |
| ..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
..... |
It is obvious that Pascal`s Triangle of the second
kind structure is built in these relations, which certainly
indicates the existing connection between Pascal`s Triangle of the
second kind columns and Lucas polynomials .
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