A136239 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).

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

Offset: 1

Author

Roger L. Bagula, Mar 16 2008

Keywords

• uned
• tabl
• sign

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 polynomials, so they are probably wrong.

Row sums are:

{1, 1, 0, -3, 4, 17, 32, -51, 0, 793}

Examples

			{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}
		

References

• page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986

Crossrefs

Cf. A137286.

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.