0,4

Comments from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 09 2008: (Start) Hochstadt defines the standard Hermite polynomials of A066325 via H(x,n+1)=x*H(x,n)-n*H(x,n-1); note the index shift relative to the definition in the current sequence.

As a consequence, the polynomials defined here are orthogonal with weight exp(-x^2/2) in a restricted sense than the usual Hermite Polynomials, i.e. the integral of P(x,n)*P(x,m)*exp(-x^2/2) over x=-infinity..infinity vanishes for m=n-1 (mod 2), as for any system of polynomials with separated even and odd functions, but not for the general case of m<>n as with the Hermite polynomials H(x,n) or other classical polynomials. (End)

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 198, pp. 8, 42-43.

P(x,0)=1; P(x,1)=x; P(x, n) = x*P(x, n - 1) - n*P(x, n - 2)

{1},

{0, 1},

{-2, 0, 1},

{0, -5, 0, 1},

{8, 0, -9, 0, 1},

{0, 33, 0, -14, 0, 1},

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

{0, -279, 0, 185, 0, -27, 0, 1},

{384, 0, -975, 0, 345, 0, -35, 0, 1},

{0, 2895, 0, -2640, 0, 588, 0, -44, 0, 1},

{-3840, 0, 12645, 0, -6090, 0, 938, 0, -54, 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]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a]

Cf. A066325.

Sequence in context: A123641 A134317 A132277 this_sequence A128890 A078924 A137526

Adjacent sequences: A137283 A137284 A137285 this_sequence A137287 A137288 A137289

sign,tabl,more

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 14 2008

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 01 2008