Toeplitz hits on Chebyshev ?

[Wouter Meeussen]

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