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Chebyshev polynomial and the Pascal
Triangle
Pafnuty Lvovich Chebyshev (May 4 1821 - November 26
1894) was a Russian mathematician. He is known for his work in the
field of probability and statistics.This article refers to what are
commonly known as Chebyshev polynomials of the first kind. Chebyshev
polynomials of the first kind are very important in numerical
approximation.
The Chebyshev polynomials can be defined as the
solution to the Chebyshev differential equation:
The Chebyshev polynomials can be defined by
trigonometric equation:
Alternatively they can be defined via the recurrence
relation:
The first few polynomials are:
See table below :
| Chebyshev Polynomials |
| 1 |
|
|
|
|
|
|
|
|
|
| 0 |
1 |
|
|
|
|
|
|
|
|
| -1 |
0 |
2 |
|
|
|
|
|
|
|
| 0 |
-3 |
0 |
4 |
|
|
|
|
|
|
| 1 |
0 |
-8 |
0 |
8 |
|
|
|
|
|
| 0 |
5 |
0 |
-20 |
0 |
16 |
|
|
|
|
| -1 |
0 |
18 |
0 |
-48 |
0 |
32 |
|
|
|
| 0 |
-7 |
0 |
56 |
0 |
-112 |
0 |
64 |
|
|
| 1 |
0 |
-32 |
0 |
160 |
0 |
-256 |
0 |
128 |
|
| 0 |
9 |
0 |
-120 |
0 |
432 |
0 |
-576 |
0 |
256 |
| T6(x)= |
-1x0 |
+0x1 |
+18x2 |
+0x3 |
-48x4 |
+0x5 |
+32x6 |
Or:
| 1/2* |
1* |
2* |
4* |
8* |
16* |
32* |
64* |
128* |
256* |
| |
|
|
|
|
|
|
|
|
|
| 2 |
|
|
|
|
|
|
|
|
|
| |
1 |
|
|
|
|
|
|
|
|
| -2 |
|
1 |
|
|
|
|
|
|
|
| |
-3 |
|
1 |
|
|
|
|
|
|
| 2 |
|
-4 |
|
1 |
|
|
|
|
|
| |
5 |
|
-5 |
|
1 |
|
|
|
|
| -2 |
|
9 |
|
-6 |
|
1 |
|
|
|
| |
-7 |
|
14 |
|
-7 |
|
1 |
|
|
| 2 |
|
-16 |
|
20 |
|
-8 |
|
1 |
|
| |
9 |
|
-30 |
|
27 |
|
-9 |
|
1 |
It is obvious that Pascal Triangle of the Second
kind structure is built in these relations, which certainly
indicates the existing connection between the Pascal Triangle of the
Second Kind and the Chebyshev polynomials of the first kind.
The Chebyshev polynomials of the second kind are
denoted  , and implemented in Mathematica as
Chebyshev U[n, x].The Chebyshev polynomials
can be defined as the solution to the Chebyshev differential
equation:
(1-x2)y''
- 3xy' + n( n+2 )y=0
The first few Chebyshev polynomials of the second
kind are :
 |
 |
 |
(1) |
 |
|
 |
(2) |
 |
|
 |
(3) |
 |
|
 |
(4) |
 |
|
 |
(5) |
 |
|
 |
(6) |
 |
|
 |
(7) |
| Chebyshev Polynomials U_n |
| 1 |
|
|
|
|
|
|
|
|
|
| 0 |
2 |
|
|
|
|
|
|
|
|
| -1 |
0 |
4 |
|
|
|
|
|
|
|
| 0 |
-4 |
0 |
8 |
|
|
|
|
|
|
| 1 |
0 |
-12 |
0 |
16 |
|
|
|
|
|
| 0 |
6 |
0 |
-32 |
0 |
32 |
|
|
|
|
| -1 |
0 |
24 |
0 |
-80 |
0 |
64 |
|
|
|
| 0 |
-8 |
0 |
80 |
0 |
-192 |
0 |
128 |
|
|
| 1 |
0 |
-40 |
0 |
240 |
0 |
-448 |
0 |
256 |
|
| 0 |
10 |
0 |
-160 |
0 |
672 |
0 |
-1024 |
0 |
512 |
| U6(x)= |
-1x0 |
+0x1 |
+24x2 |
+0x3 |
-80x4 |
+0x5 |
+64x6 |
Or:
| 1* |
2* |
4* |
8* |
16* |
32* |
64* |
128* |
256* |
512* |
| |
|
|
|
|
|
|
|
|
|
| 1 |
|
|
|
|
|
|
|
|
|
| |
1 |
|
|
|
|
|
|
|
|
| -1 |
|
1 |
|
|
|
|
|
|
|
| |
-2 |
|
1 |
|
|
|
|
|
|
| 1 |
|
-3 |
|
1 |
|
|
|
|
|
| |
3 |
|
-4 |
|
1 |
|
|
|
|
| -1 |
|
6 |
|
-5 |
|
1 |
|
|
|
| |
-4 |
|
10 |
|
-6 |
|
1 |
|
|
| 1 |
|
-10 |
|
15 |
|
-7 |
|
1 |
|
| |
5 |
|
-20 |
|
21 |
|
-8 |
|
1 |
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 Chebyshev
polynomials of the second kind.
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