1,4

Row sums:

{1, 1, 0, -3, -18, -129, -1182, -13281, -176478, -2704119, -46909362}

The relationship between the Bessel polynomials and the Hermite

polynomials is better shown by this set of functions.

It took me a while to figure how to do this right: this result is like a toral inverse (1/x) of the Bessel polynomials.

Reversing the coefficient vectors and doing a Gram-Schmidt on them:

f[n_]:=Table[0,{i,1,n}];

b=Table[Join[a[[n]],f[Length[a]-n]],{n,1,Length[a]}];

Table[Length[b[[n]]],{n,1,Length[b]}];

<<LinearAlgebra`Orthogonalization`;

c=GramSchmidt[b];

Gives that these polynomials are at best alternating orthogonal.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972, 10th edition, (and various reprintings), p. 631.

p(x,0)=1;p(x,1)=1/x; P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2); with substitution of 1/y for x.

{1},

{0, 1},

{-2, 0, 2},

{0, -11, 0, 8},

{8, 0, -74, 0, 48},

{0, 119, 0, -632, 0, 384},

{-48, 0,1634, 0, -6608, 0, 3840},

{0, -1409, 0, 24032, 0, -81984, 0, 46080},

P[x, 0] = 1; P[x, 1] = 1/x; P[x_, n_] := P[x, n] = 2*(n-1)*P[x, n - 1]/x - n*P[x, n - 2]; Table[ExpandAll[P[x, n] /. x -> 1/y], {n, 0, 10}]; a = Table[CoefficientList[P[x, n] /. x -> 1/y, y], {n, 0, 10}]; Flatten[a]

Cf. A106174, A123956.

Adjacent sequences: A136665 A136666 A136667 this_sequence A136669 A136670 A136671

Sequence in context: A099554 A107729 A113400 this_sequence A057498 A137949 A019214

uned,tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 03 2008