A137561 A triangular sequence of coefficients of the fixed point Chebyshev polynomials: p(x,n)=T(x,n)-x:A053120[x,n]-x.

1, -1, 0, -1, -1, 2, 0, -4, 0, 4, 1, -1, -8, 0, 8, 0, 4, 0, -20, 0, 16, -1, -1, 18, 0, -48, 0, 32, 0, -8, 0, 56, 0, -112, 0, 64, 1, -1, -32, 0, 160, 0, -256, 0, 128, 0, 8, 0, -120, 0, 432, 0, -576, 0, 256, -1, -1, 50, 0, -400, 0, 1120, 0, -1280, 0, 512

Offset: 1

Author

Roger L. Bagula, Apr 25 2008

Keywords

• tabl
• uned
• sign

Comments

The row sums are all zero.

The idea of roots of polynomials of the sort came from the realization that in Umbral calculus for the expansion function:

p(x,t)=Sum(P(xd,n)*t^n/n!,{n,0,Infinity}];

to actually work there has to be a convergent limit:

Limit[P(x,n)*t^n/n!,n->Infinity]=0;

The idea that a point gets "trapped" in complex dynamics is the iterative:

Pc[x,n]=x

So if we look at polynomials as iterative steps, at a fixed point

the roots would be important dynamically.

Examples

			{1, -1},
{0},
{-1, -1, 2},
{0, -4, 0, 4},
{1, -1, -8, 0, 8},
{0, 4, 0, -20, 0, 16},
{-1, -1,18, 0, -48, 0, 32},
{0, -8, 0, 56, 0, -112, 0, 64},
{1, -1, -32, 0, 160, 0, -256, 0, 128},
{0, 8, 0, -120, 0, 432, 0, -576, 0, 256},
{-1, -1,50, 0, -400, 0, 1120, 0, -1280, 0, 512}
		

References

• Lennart Carleson, Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993,Chapter II, page 27 ff

Programs

Mathematica

Table[ChebyshevT[n, x] - x, {n, 0, 10}]; a = Table[CoefficientList[ChebyshevT[n, x] - x, x], {n, 0, 10}]; Flatten[{{1, -1}, {0}, {-1, -1, 2}, {0, -4, 0, 4}, {1, -1, -8, 0, 8}, {0, 4, 0, -20, 0, 16}, {-1, -1, 18, 0, -48, 0, 32}, {0, -8, 0, 56, 0, -112, 0, 64}, {1, -1, -32, 0, 160, 0, -256, 0, 128}, {0, 8, 0, -120, 0, 432, 0, -576, 0, 256}, {-1, -1, 50, 0, -400, 0, 1120, 0, -1280, 0, 512}}]

Formula

p(x,n)=T(x,n)-x:A053120[x,n]-x; out_n,m=Coefficients(A053120[x,n]-x).