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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| 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 polynomial, so they are probably wrong.
Row sums are:
{1, 1, 0, -3, 4, 17, 32, -51, 0, 793}
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REFERENCES
| page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
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FORMULA
| 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
| {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
| Cf. A137286.
Sequence in context: A157391 A099097 A152150 * A058175 A112906 A137375
Adjacent sequences: A136236 A136237 A136238 * A136240 A136241 A136242
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 16 2008
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