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Special Polynomials
(2012-02-15)
Chebyshev Polynomials [ of the first kind ]
A family of commuting polynomial functions.
Tn oTp
= Tp oTn
= Tnp
cos(nq) is a polynomial
function of cos(q).
The following relation defines a polynomial of degree n
known as the Chebyshev polynomial of degree n:
cos (nq) =
Tn (cos q)
The symbol T comes from transliterations, like Tchebycheff
or Tchebychev, which are better matches for the Russian pronounciation
(the spellings "Chebychev" and "Tchebyshev" also appear).
The trigonometric formula
cos(n+2)x = 2 cos x cos(n+1)x - cos nx
translates into a simple recurrence relation which makes Chebyshev polynomials
very easy to tabulate: Tn+2(x) =
2x Tn+1(x) - Tn(x)
| T0(x) | = |
1 |
| T1(x) | = |
| x |
| T2(x) | = |
-1 | | +2x2 |
| T3(x) | = |
| -3x | | +4x3 |
| T4(x) | = |
1 | | -8x2 | | +8x4 |
| T5(x) | = |
| 5x | | -20x3 | | +16x5 |
| T6(x) | = |
-1 | | +18x2 | | -48x4 |
| +32x6 |
| T7(x) | = |
| -7x | | +56x3 | | -112x5 |
| +64x7 |
| T8(x) | = |
1 | | -32x2 | | +160x4 |
| -256x6 | | +128x8 |
Chebyshev Economization
(2012-02-15)
Legendre Polynomials
The key to Coulombian multipole expansion
and spherical harmonics.
The Legendre polynomials
(A008316)
are:
| P0 (x) | = |
1 |
|
Pn(x) = (2-1/n) x Pn-1(x)
-
(1-1/n) Pn-2(x)
|
| P1 (x) | = |
| x | |
 |
| 2 | P2 (x) | = |
-1 | + | 3 x2 |
|
| 2 | P3 (x) | = |
| -3 x | + | 5 x3 |
| 8 | P4 (x) | = |
3 | - | 30 x2 |
+ | 35 x4 |
| 8 | P5 (x) | = |
| 15 x | - | 70 x3 |
+ | 63 x5 |
| 16 | P6 (x) | = |
-5 | + | 105 x2 |
- | 315 x4 |
+ | 231 x6 |
| 16 | P7 (x) | = |
| -35 x | + | 315 x3 |
- | 693 x5 |
+ | 429 x7 |
They are linked to the expressions of spherical harmonics
in terms of the colatitude
q Î [0,p[ and the
longitude
f (modulo
2p).
MathWorld :
Legendre Polynomials
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Spherical Harmonics
(2012-02-16)
Laguerre Polynomials
Radial part of the solution of the Schrödinger equation
for hydrogenoids.
| | L0(x) | = |
1 |
(n+1) Ln+l (x) = (2n+1-x) Ln (x) - n Ln-1 (x) |
| | L1(x) | = |
1 | - x |
| 2 | L2(x) | = |
2 | - 4x | + x2 |
| 6 | L3(x) | = |
6 | - 18x | + 9x2 | - x3 |
| 24 | L4(x) | = |
24 | - 96x | + 72x2 | - 16x3 | + x4 |
| 120 | L5(x) | = |
120 | - 600x | + 600x2 | - 200x3 | + 25x4 | + x5 |
| 720 | L6(x) | = |
720 | - 4320x | + 5400x2 | - 2400x3 | + 450x4 |
- 36x5 | + x6 |
| 5040 | L7(x) | = |
5040 | -35280x | +52920x2 | -29400x3 | +7350x4 | -882x5 |
+49x6 | -x7 |
Edmond Laguerre |
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Edmond Laguerre (1834-1886) may have devised those polynomials as early as 1860
but the relevant memoir was only published in 1879.
The Laguerre polynomials arose from a remarkable
continued fraction expansion
of the definite integral from zero to infinity of exp(x)/x
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Wikipedia :
Laguerre Polynomials
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Rook Polynomials
of John Riordan (1903-1988)
MathWorld :
Laguerre Polynomials
| MacTutor :
Edmond
Laguerre (1834-1886; X1853)
(2012-02-18)
Hermite Polynomials & Modified Hermite Polynomials
Eigenstates of the quantum harmonic oscillator.
| H0 (x) | = |
1 |
|
Hn+1(x) = 2x Hn(x)
-
2n Hn-1(x)
|
| H1 (x) | = |
| 2 x | |
|
| H2 (x) | = |
-2 | + | 4 x2 |
|
| H3 (x) | = |
| -12 x | + | 8 x3 |
| H4 (x) | = |
12 | - | 48 x2 |
+ | 16 x4 |
| H5 (x) | = |
| 120 x | - | 160 x3 |
+ | 32 x5 |
| H6 (x) | = |
-120 | + | 720 x2 |
- | 480 x4 |
+ | 64 x6 |
| H7 (x) | = |
| -1680 x | + | 3360 x3 |
- | 1344 x5 |
+ | 128 x7 |
The above are more popular than the
modified Hermite polynomials Hen which can be defined via:
Hn (x) = 2n/2 Hen (2½ x)
| He0 (x) | = |
1 |
|
Hen+1(x) = x Hen(x)
-
n Hen-1(x)
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| He1 (x) | = |
| x | |
|
| He2 (x) | = |
-1 | + | x2 |
|
| He3 (x) | = |
| -3 x | + | x3 |
| He4 (x) | = |
3 | - | 6 x2 |
+ | x4 |
| He5 (x) | = |
| 15 x | - | 10 x3 |
+ | x5 |
| He6 (x) | = |
-15 | + | 45 x2 |
- | 15 x4 |
+ | x6 |
| He7 (x) | = |
| -105 x | + | 105 x3 |
- | 21 x5 |
+ | x7 |
Hermite Polynomials (Wikipedia)
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Hermite Polynomials (MathWorld)
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