A136586 Triangle of coefficients of even modified recursive orthogonal Hermite polynomials given in Hochstadt's book:P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) ;A137286; P2(x,n)=P(x,n)+P(x,n-2).

0, 0, 1, -1, 0, 1, 0, -4, 0, 1, 6, 0, -8, 0, 1, 0, 28, 0, -13, 0, 1, -40, 0, 78, 0, -19, 0, 1, 0, -246, 0, 171, 0, -26, 0, 1, 336, 0, -888, 0, 325, 0, -34, 0, 1, 0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1, -3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1

Offset: 1

Author

Roger L. Bagula, Mar 30 2008

Keywords

• uned
• tabl
• sign

Comments

Row sums are:

{0, 1, 0, -3, -1, 16, 20, -100, -260, 680, 3320}

The double function Integration is alternating:

Table[Integrate[Exp[ -x^2/2]*P2[x, n]*P2[x, m], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0, 10}];

Four Initial conditions were necessary for starting this recursion:

P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1;

Examples

			{0},
{0, 1},
{-1, 0, 1},
{0, -4, 0, 1},
{6, 0, -8, 0, 1},
{0, 28, 0, -13, 0, 1},
{-40, 0, 78, 0, -19, 0, 1},
{0, -246, 0, 171, 0, -26, 0,1},
{336, 0, -888, 0, 325, 0, -34, 0, 1},
{0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1},
{-3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1}
		

Crossrefs

Cf. A137286.

Programs

Mathematica

P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = P[x, n] + P[x, n - 2]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Join[{0}, Table[CoefficientList[P2[x, n], x], {n, 0, 10}]]; Flatten[a]

Formula

H2(x,n)=A137286(x,n)+A137286(x,n-2)