_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Mar 30 2008

nonn,uned,tabl,newsign

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,2*n): second type.

2, -2, 1, 1, 6, 0, -8, 0, 1, -48, -5, 87, 1, -20, 0, 1, 392, 0, -984, 0, 346, 0, -35, 0, 1, -3840, 33, 12645, -14, -6090, 1, 938, 0, -54, 0, 1, 46032, 0, -187338, 0, 114745, 0, -23813, 0, 2070, 0, -77, 0, 1, -645120, -279, 3133935, 185, -2336040, -27, 611415, 1, -71280, 0, 3993, 0, -104, 0, 1, 10322304, 0, -58438830, 0

1,1

Row sums are:

{2, 0, -1, 16, -280, 3620, -48380, 696680, -10740280, 175631200, -3000871600}

The double function Integration is not orthogonal:

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

Both types have the polynomial:

p[x]=6 - 8 x^2 + x^4

Original Hermite has:

p[x]=8 - 9 x^2 + x^4

Integration of these as:

f[y_] = Integrate[Exp[ -x^2/4]*p[x], {x, Infinity, y}]

g[z_] = Integrate[Exp[ -y^2/4]*f[y], {y, Infinity, z}]

gives three level or four level curves with negative limit of 2*Pi.

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

{2},

{-2, 1, 1},

{6, 0, -8, 0, 1},

{-48, -5, 87,1, -20, 0, 1},

{392, 0, -984, 0, 346, 0, -35, 0, 1},

{-3840, 33, 12645, -14, -6090, 1, 938, 0, -54, 0, 1},

{46032, 0, -187338, 0,114745, 0, -23813, 0, 2070, 0, -77, 0, 1},

{-645120, -279, 3133935, 185, -2336040, -27, 611415, 1, -71280, 0,3993, 0, -104,0, 1},

{10322304, 0, -58438830, 0, 51450870, 0, -16289000, 0, 2386396, 0, -178893, 0, 7007, 0, -135, 0, 1},

{-185794560, 2895, 1203216525, -2640, -1223803350,

588, 455259420, -44, -80424630, 1, 7561554, 0, -395850, 0, 11460, 0, -170,

0, 1},

{3715887360, 0, -27125479980, 0, 31335461535, 0, -13408093762, 0, 2775672846, 0,-314143829, 0, 20603310, 0, -796620, 0, 17748, 0, -209, 0, 1}

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

Cf. A137286.

nonn,uned,tabl

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 30 2008

approved