new procedure for converting a new recursive polynomial set into matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg70726] new procedure for converting a new recursive polynomial set into matrices
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Tue, 24 Oct 2006 02:24:58 -0400 (EDT)

```This procedure connects matrix representations to combinatorial
trangular sequences in a direct manner!

Here are the matrices my Mathematica produces:
2by2
{{0, 1},
{1, -1}}
3by3
{{0, 1, 0},
{0, 0, 1},
{0, 2, -1}}
4by4
{{0, 1, 0, 0},
{0, 0, 1, 0},
{0, 0, 0, 1},
{-2, -2, 4, -1}}
5by5
{{0, 1, 0, 0, 0},
{0, 0, 1, 0, 0},
{0, 0, 0, 1, 0},
{0, 0, 0, 0, 1},
{6, -2, -11, 7, -1}}
6by6
{{0, 1, 0, 0, 0, 0},
{0, 0, 1, 0, 0, 0},
{0, 0, 0, 1, 0, 0},
{0, 0, 0, 0, 1, 0},
{0, 0, 0, 0, 0, 1},
{-22, 24, 37, -42, 12, -1}}
How I get matrices from recursive polynomials: connections between
sequence types:
Starting with the first easy and linear type in  Fibonacci/ Bonacci
sequences:
1) Recursive transforms like
f[n]=f[n-1]+f[n-2]
are connected to vector Matrix Markovs:
2) M={{0,1}
,{1,1}}
v[1]={0,1}
v[n]=M^n.v[1]
3) characteristic polynomials of these both
give coefficient expansions: ( toral inversion forms)
f[x]=x2-x-2
g[x]=x2*f[1/x]
coefficient expansion of[ x/g[x]]
4) Solutions of the recursive transforms gives the Binet F[n] types
of sequences.
5) digraph substitution algebras also produce these sequences.
That is five different ways of doing the same thing.
Well,
I've found a similar set of  relationships in triangular sequence like
those characterized by the binomial.
1) triangular sequence as in Pascal's triangle of Combinations
2) polynomial recursive formulations whose coefficients are the
triangular sequences
3) Matrices whose characteristic polynomials are the recursive polynomials
4) row sum sequences of the triangulars
5) Matrix Markovs generated by the matrices
6) coefficient expansions of the recursive polynomials

What Gary and I've been investigating lately is the relationship of
"Field" type rules:
Steinbach suggested that the sequences associated with diagonals of
polynomials
were involved in a field like structure.
What  we have been looking at is sums and products of this basic matrix
type and how they relate to the base
matrix.
What I discovered was that if you have the polynomials ,
you can get matrices.
If you have polynomials, you can get  a triangular sequence.
If you have a triangular sequence , you can generate matrices.
I'm sure that n baesd functions like the Binets also follw this pattern
as well.

Since the matrices  are closely related to different graph and
substitution algebras as well
geomtery, differential equations  and symmetry, we have one of the most
extensinve systems of interlocking
mathematics here that is known.

{{1},
{-1, 1},
{0, -2, 1},
{2, 2, -4, 1},
{-6, 2, 11, -7, 1},
{22, -24, -37, 42, -12, 1},
{-146, 204, 217, -338, 133, -20,1},
{1766, -2654, -2395, 4359, -1995, 387, -33, 1},
{-36064, 56072,  46122, -91568, 45323, -9982, 1073, -54, 1},
{1212048, -1921280, -1492916, 3124562, -1616590, 381615, -46200, 2901,
-88, 1},

Mathematica code:
Clear[p, a, b, An]
p[-1, x] = 0; p[0, x] = 1;
p[k_, x_] := p[k, x] = (-Fibonacci[n] + x)*p[k - 1, x] - (n - 1)*p[k - 2, x]
Table[Expand[p[n, x]], {n, 0, 10}]
Table[Sum[CoefficientList[p[n, x], x][[m]], {m,
1, Length[CoefficientList[p[n, x], x]]}], {n, 0, 15}]
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]
Flatten[w]
An[d_] := Table[If[n == d, -w[[n]][[m]], If[m == n + 1, 1, 0]], {n, 1,
d}, {m, 1, d}]
Table[An[d], {d, 1, 10}]
Table[CharacteristicPolynomial[An[d], x], {d, 1, 10}]
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x],
x], {d,
1, 10}]]
Flatten[%]
Table[NSolve[CharacteristicPolynomial[An[d], x] == 0, x], {d, 2, 10}]

Roger Bagula (rlbagula at sbcglobal.net), Oct 23 2006

```

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