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A139037
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A triangular sequence of coefficients Chebyshev T(x,n) as Besicovitch-Ursell polynomials ( fractal orthogonal functions): f(x,n)=ChebyshevT[n,2^n*x]/(2^*s*n).
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0
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1, 0, 2, -1, 0, 32, 0, -24, 0, 2048, 1, 0, -2048, 0, 524288, 0, 160, 0, -655360, 0, 536870912, -1, 0, 73728, 0, -805306368, 0, 2199023255552, 0, -896, 0, 117440512, 0, -3848290697216, 0, 36028797018963968, 1, 0, -2097152, 0, 687194767360, 0, -72057594037927936, 0, 2361183241434822606848, 0, 4608, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums are:
{1, 2, 31, 2024, 522241, 536215712, 2198218022911, 36024948845706368,
2361111184527977349121, 618964706995596541734949376,
649035559893095618486323487178751};
The integration of these functions is alternating orthogonal as are a lot of secondary Chebyshev polynomial sets:
TableForm[Table[Integrate[(ChebyshevT[n, 2^n*x]/2^(s*n))*(ChebyshevT[m, 2^m*x]/2^(s*m))/Sqrt[1 -x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]];
The fractal dimensional scaling here acts as a constant during integration.
These "biscuit" type functions are closely related to Weierstrass fractal
and are usually constructed with a unit square "cartoon".
Graphing the polynomials shows that it is the "even" functions that are orthogonal. Orthogonal fractal systems are found in quantum fractals.
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REFERENCES
| Authors?, http://www.cns.gatech.edu/~danek/preprints/prl.85.5022.pdf, Time Evolution of Quantum Fractals, Phys. Rev. Lett. 85, 5022 - 5025 (2000)
G. A. Edgar, Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990, 202-206.
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FORMULA
| s=Log[2]/Log[3]; f(x,n)=ChebyshevT[n,2^n*x]/(2^*s*n); out_n,m=(2^*s*n)*Coefficients(f(x,n))
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EXAMPLE
| {1},
{0, 2},
{-1, 0, 32},
{0, -24, 0, 2048},
{1, 0, -2048,0, 524288},
{0, 160, 0, -655360, 0, 536870912},
{-1, 0, 73728, 0, -805306368, 0, 2199023255552},
{0, -896, 0, 117440512, 0, -3848290697216, 0, 36028797018963968},
{1, 0, -2097152, 0, 687194767360, 0, -72057594037927936, 0, 2361183241434822606848},
{0, 4608, 0, -16106127360, 0, 15199648742375424, 0, -5312662293228350865408, 0, 618970019642690137449562112},
{-1, 0, 52428800, 0, -439804651110400, 0,1291272085159668613120, 0, -1547425049106725343623905280,0, 649037107316853453566312041152512}
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MATHEMATICA
| s = Log[2]/Log[3]; g = Table[ChebyshevT[n, 2^n*x]/2^(s*n), {n, 0, 10}]; a = Table[CoefficientList[ChebyshevT[n, 2^n*x], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[ChebyshevT[n, 2^n*x], x]], {n, 0, 10}];
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CROSSREFS
| Sequence in context: A202697 A051652 A077019 * A108511 A196272 A086073
Adjacent sequences: A139034 A139035 A139036 * A139038 A139039 A139040
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), May 31 2008
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