Toeplitz hits on Chebyshev ?
Wouter Meeussen
eu000949 at pophost.eunet.be
Mon Sep 18 20:10:40 CEST 2000
the characteristic equation of the Toeplitz matrix
Table[ i + j, {i, 0, n}, {j, 0, n}]
or, nicer :
0 1 2 ... n
1 2 3 ... n+1
2 3 4 ... n+2
...
n n+1 ... 2n
is, for n=1..8 :
{-1 - 2*x + x^2,
6*x + 6*x^2 - x^3,
-20*x^2 - 12*x^3 + x^4,
50*x^3 + 20*x^4 - x^5,
-105*x^4 - 30*x^5 + x^6,
196*x^5 + 42*x^6 - x^7,
-336*x^6 - 56*x^7 + x^8,
540*x^7 + 72*x^8 - x^9,
...
the first column being
-1, 6, -20, 50, -105, 196, -336, 540
which hits on A002415 or A008057,
given as n^2*(n^2-1)/12 (for n starting at 2)
or (n-1)*binomial(n,3)/2 (for n starting at 3)
Both sequences are linked to the Chebyshev polynomials
was it to be expected that Toeplitz hits on Chebyshev?
wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be
More information about the SeqFan
mailing list