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A136239
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Forced end points ( -Infinity ->-1) integration of A137286: Triangle of coefficients of Integrated recursive orthogonal Hermite polynomials given in Hochstadt's book : P(x, n) = x*P(x, n - 1) - n*P(x, n - 2).
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1
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1, 0, 1, -1, 0, 1, -1, -3, 0, 1, 9, 0, -6, 0, 1, -1, 27, 0, -10, 0, 1, -19, 0, 65, 0, -15, 0, 1, -1, -165, 0, 135, 0, -21, 0, 1, 399, 0, -624, 0, 252, 0, -28, 0, 1, -1, 2145, 0, -1750, 0, 434, 0, -36, 0, 1
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OFFSET
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1,8
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COMMENTS
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Because of error functions in the result where constants should be this is a difficult calculation.
Probably the wrong approach, but it is my best effort at getting Gaussian normal type functions to give integers. There has got to be a better way than this: maybe a conformal transform of the known Chebyshev Integration polynomials?
No recurrence formula was found for these polynomials, so they are probably wrong.
Row sums are:
{1, 1, 0, -3, 4, 17, 32, -51, 0, 793}
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REFERENCES
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page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
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LINKS
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FORMULA
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P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); L(x,n)=Integrate[Exp[y^2/4]*p(y,n-1),{y,-Infinity,x}]/(-2*Exp[ -x^2/4]) Here the weight function is taken as the square root of the Hermite weight function Exp[ -x^2/2] and then divided out of the end result.
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EXAMPLE
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{1},
{0, 1},
{-1, 0, 1},
{-1, -3, 0, 1},
{9, 0, -6, 0, 1},
{-1, 27, 0, -10, 0, 1},
{-19, 0, 65, 0, -15, 0, 1},
{-1, -165, 0, 135, 0, -21, 0,1},
{399, 0, -624, 0, 252, 0, -28, 0, 1},
{-1, 2145, 0, -1750, 0, 434, 0, -36, 0, 1}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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