Intended for: November 15, 2012
Timetable
- First draft entered by Wolfdieter Lang on November 6, 2011 ✓
- Draft reviewed by Daniel Forgues on October 16, 2016 (presentation review) (help needed: may some editor please review the content) ✓
- Draft to be approved by October 15, 2012
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A049310: Triangle of coefficients of Chebyshev ’s
polynomials* (exponents in increasing order).
-
{ 1, 0, 1, −1, 0, 1, 0, −2, 0, 1, ... }
This sequence appears in linear atomic chains with
uniformly harmonic interacting atoms of the same mass. The eigenmodes have scaled frequency squares
given by the zeros of
.
The recurrence for the oscillations with frequency
and displacement
from the equilibrium position at site no.
,
(
being the complex unit) is
-
qn + 1 − 2 (1 − x) qn + qn − 1 = 0. |
Here we have
, the normalized frequency squared, with
, where
is the uniform spring constant and
is the atom’s mass. This leads to a so called
transfer matrix
with (
for matrix transpose)
-
[qn +1, qn ] T = R (x) [qn , qn − 1] T. |
Iteration yields
-
[qn +1, qn ] T = M n(x) [q1, q0 ] T, |
with
, and the two arbitrary inputs
and
. It follows that
M n(x) = | | | | S (n, 2 (1 − x)) | | S (n − 1, 2 (1 − x)) |
− S (n − 1, 2 (1 − x)) | | − S (n − 2, 2 (1 − x)) |
| | | | |
|
due to the recurrence for the
-polynomials:
-
S (−1, x) = 0; S (0, x) = 1; S (n, x) = x S (n − 1, x) − S (n − 2, x), n ≥ 1. |
Thus one obtains the general solution for the displacements
-
qn +1(x) = S (n, 2 (1 − x)) q1 − S (n − 1, 2 (1 − x)) q0 . |
For finite
-chains, with fixed boundary conditions
, one therefore has to solve
, and thus obtains the
normalized eigenfrequency squares for the
-chain:
-
x ( N ) k = 2 sin 2, k = 1, ..., N. |
A side remark: because
, also
, identically, therefore on has the so called Cassini–Simson identity
-
(S (n − 1, y)) 2 − S (n, y) S (n − 2, y) = 1, n ≥ 0. |
For this and another nine applications of this sequence and the row polynomials
see the a link under
A049310.
_______________
* Chebyshev polynomials