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Re: new procedure for converting a new recursive polynomial set into matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70908] Re: new procedure for converting a new recursive polynomial set into matrices
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Wed, 1 Nov 2006 03:55:49 -0500 (EST)
  • References: <ehkcmn$jn9$1@smc.vnet.net>

I made up a version much like a classic tridiagonal
( 2's center -1's off diagonal as -2's center 1's off diagonal)
Recursive polynomial version: A078812
b[n_] = -2; a[n_] = 1;
p[0, x] = 1; p[1, x] = (x - b[1])/a[1];
p[k_, x_] := p[k, x] = ((x - b[n - 1])*p[k - 1, x] - a[n - 2] *
p[k - 2, x])/a[n - 1]
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]

Matrix version:
T[n_, m_] := If[ n == m, -2, If[n == m - 1 || n == m + 1, 1, 0]]
M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]
a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], 
x], {d, 1, 10}]]




%I A000001
%S A000001 -2, -2, -1, 3, 4, 1, -4, -10, -6, -1, 5, 20, 21, 8, 1, -6, 
-35, -56, -36,
-10, -1, 7, 56, 126, 120, 55, 12, 1, -8, -84, -252, -330, -220, -78, 
-14, -1,
9, 120, 462, 792, 715, 364, 105, 16, 1, -10, -165, -792, -1716, -2002, 
-1365,
-560, -136, -18, -1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1
%N A000001 tridiagonal matrix version of A078812
%C A000001 It seems very important to me that these two ways of 
representing the data are essentually equivalent. Except for the first 
element and signs they are the same.
Recursive polynomial version in Mathematica:
b[n_] = -2; a[n_] = 1;
p[0, x] = 1; p[1, x] = (x - b[1])/a[1];
p[k_, x_] := p[k, x] = ((x - b[n - 1])*p[k - 1, x] - a[n - 2] *p[k - 2, 
x])/a[n - 1];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]
Triangular sequence:
A078812 in OEIS
{1},
{2, 1},
{3, 4, 1},
{4, 10, 6, 1},
{5, 20, 21, 8, 1},
{6, 35, 56, 36, 10, 1},
{7, 56, 126, 120, 55, 12, 1},
{8, 84, 252, 330, 220, 78, 14, 1},
{9, 120, 462, 792, 715, 364, 105, 16, 1},
{10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1},
{11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1}
%D A000001 Joanne Dombrowski,
Tridiagonal matrix representations of cyclic selfadjoint operators,
Pacific J. Math. 114, no. 2 (1984), 325?334
%F A000001 a(n,m)=If[ n == m, -2, If[n == m - 1 || n == m + 1, 1, 0]]
%e A000001 Triangular sequence:
{-2},
{-2, -1},
{3, 4,1},
{-4, -10, -6, -1},
{5, 20, 21, 8, 1},
{-6, -35, -56, -36, -10, -1},
{7, 56, 126, 120, 55, 12, 1},
{-8, -84, -252, -330, -220, -78, -14, -1},
{9, 120, 462, 792, 715, 364, 105, 16, 1},
{-10, -165, -792, -1716, -2002, -1365, -560, -136, -18, -1},
{11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1}
Matrices:
{{-2}},

{{-2, 1},
{1, -2}},

{{-2, 1, 0},
{1, -2, 1},
{0,1, -2}},

{{-2, 1, 0, 0},
{1, -2, 1, 0},
{0, 1, -2, 1},
{0, 0, 1, -2}},

{{-2, 1, 0, 0, 0},
{1, -2, 1, 0, 0},
{0, 1, -2, 1, 0},
{0, 0, 1, -2, 1},
{0, 0, 0, 1, -2}}
%t A000001 T[n_, m_] := If[ n == m, -2, If[n == m - 1 || n == m + 1, 1, 0]]
M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]
Table[M[d], {d, 1, 10}]
Table[Det[M[d]], {d, 1, 10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]
a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {
d, 1, 10}]];
Flatten[a]
%Y A000001 Cf. A078812
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger Bagula and Gary Adamson (rlbagula at sbcglobal.net), Oct 
30 2006

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