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 A139360 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. 1

%I #6 Nov 04 2013 02:21:54

%S 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,

%T 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,

%U 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

%N 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.

%C All the row sums are 3.

%C These polynomials are two level triangles:

%C m levels and n levels.

%C The integration table is:

%C TableForm[Table[Integrate[p[x, n, m]/Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]]

%C As binomials these polynomials are the quantum mechanics of a type of 2 dimensional crystal that vibrates much like a Chladni standing wave.

%C They come from thinking of Chebyshev polynomials in terms of a Ring structure in a commutative algebra.

%D 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.

%D Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press. New York, 2007, p. 237.

%D Advanced Number Theory, Harvey Cohn, Dover Books, 1963, p. 114.

%F 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).

%e {{1, 2}},

%e {{-1, 0, 4}, {-1, 0, 2, 2}},

%e {{1, -6, 0, 8}, {0, -2, -3, 4, 4}, {-1,0, 2, -6, 0, 8}},

%e {{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}},

%e {{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}},

%e {{-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}}

%t 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}]]

%K tabf,uned,sign

%O 1,2

%A _Roger L. Bagula_, Jun 08 2008

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Last modified September 26 21:54 EDT 2023. Contains 365668 sequences. (Running on oeis4.)