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A139360
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Coefficients of Chebyshev T(x,n) constructed polynomials that are based on a Cumrun Vafa type of Calabi-Yau crystal (binomials of this type would behave much like Chladni standing waves): p(x,n,m)=T(x,n)*T(x,m)+T(x,n)+T(x,m): binomials are: p(x,y,n,m)=T(x,n)*T(y,m)+T(x,n)+T(y,m). Integrate(p(x,n,m)/sqrt[1-x^2),{x,-1,1}]=0 if n,m>0 and n does not equal m: for n=m the result is Pi/2: they are orthogonal polynomials.
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1
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1, 2, -1, 0, 4, -1, 0, 2, 2, 1, -6, 0, 8, 0, -2, -3, 4, 4, -1, 0, 2, -6, 0, 8, 3, 0, -16, 0, 16, 1, 2, -8, -8, 8, 8, -1, 0, 4, 0, -16, 0, 16, 1, -6, -8, 32, 8, -56, 0, 32, 1, 10, 0, -40, 0, 32, 0, 6, 5, -20, -20, 16, 16, -1, 0, 2, 10, 0, -40, 0, 32, 0, 2, -15, -16, 80, 16, -128, 0, 64, 1, 10, -8, -80, 8, 232, 0, -288, 0, 128, -1, 0, 36, 0, -96, 0
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OFFSET
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1,2
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COMMENTS
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All the row sums are 3.
These polynomials are two level triangles:
m levels and n levels.
The integration table is:
TableForm[Table[Integrate[p[x, n, m]/Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]]
As binomials these polynomials are the quantum mechanics of a type of 2 dimensional crystal that vibrates much like a Chladni standing wave.
They come from thinking of Chebyshev polynomials in terms of a Ring structure in a commutative algebra.
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REFERENCES
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D-branes as defects in the Calabi-Yau crystal. Natalia Saulina, Cumrun Vafa (Harvard U., Phys. Dept.). HUTP-04-A018, Apr 2004. 28pp. e-Print: hep-th/0404246.
Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press. New York, 2007, p. 237.
Advanced Number Theory, Harvey Cohn, Dover Books, 1963, p. 114.
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LINKS
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FORMULA
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p(x,n,m)=T(x,n)*T(x,m)+T(x,n)+T(x,m): For m<n: out_n,m=Coefficients(P(x,n,m).
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EXAMPLE
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{{1, 2}},
{{-1, 0, 4}, {-1, 0, 2, 2}},
{{1, -6, 0, 8}, {0, -2, -3, 4, 4}, {-1,0, 2, -6, 0, 8}},
{{3, 0, -16, 0, 16}, {1, 2, -8, -8, 8, 8}, {-1, 0, 4, 0, -16, 0, 16}, {1, -6, -8, 32, 8, -56, 0, 32}},
{{1, 10, 0, -40, 0, 32}, {0, 6, 5, -20, -20, 16, 16}, {-1, 0, 2, 10, 0, -40, 0, 32}, {0, 2, -15, -16, 80, 16, -128, 0, 64}, {1, 10, -8, -80, 8, 232, 0, -288, 0, 128}},
{{-1, 0, 36, 0, -96, 0, 64}, {-1, 0, 18, 18, -48, -48, 32, 32}, {-1, 0, 0, 0, 36, 0, -96, 0, 64}, {-1, 0, 18, -54, -48, 216, 32, -288, 0, 128}, {-1, 0, 36, 0, -240, 0, 592, 0, -640, 0, 256}, {-1, 0, 18,90, -48, -600, 32, 1408, 0, -1408, 0, 512}}
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MATHEMATICA
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Clear[p, a] p[x_, n_, m_] := ChebyshevT[n, x]*ChebyshevT[m, x] + ChebyshevT[n, x] + ChebyshevT[m, x]; Table[Table[ExpandAll[p[x, n, m]], {m, 0, n - 1}], {n, 0, 10}]; a = Table[Table[CoefficientList[p[x, n, m], x], {m, 0, n - 1}], {n, 0, 10}]; Flatten[a] Flatten[Table[Table[Apply[Plus, CoefficientList[p[x, n, m], x]], {m, 0, n - 1}], {n, 0, 10}]]
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CROSSREFS
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KEYWORD
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tabf,uned,sign
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AUTHOR
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STATUS
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approved
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