A137286 - OEIS

A137286

Triangle of coefficients of a version of the Hermite polynomials defined by 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

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OFFSET

0,4

COMMENTS

From R. J. Mathar, 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)

REFERENCES

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

LINKS

Table of n, a(n) for n=0..65.

R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 333 and A066325/A099174. From Tom Copeland, Jan 03 2016)

FORMULA

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

EXAMPLE

{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}

MATHEMATICA

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]

PROG

(PARI) polx(n) = if (n == 0, 1, if (n == 1, x, x*polx(n - 1) - n*polx(n - 2)));

tabl(nn) = {for (n = 0, nn, pol = polx (n); for (i = 0, n, print1(polcoeff(pol, i), ", "); ); print(); ); } \\ Michel Marcus, Feb 12 2014

(Python)

from sympy import Poly

from sympy.abc import x

def P(x, n): return 1 if n==0 else x if n==1 else x*P(x, n - 1) - n*P(x, n - 2)

def a(n): return Poly(P(x, n), x).all_coeffs()[::-1]

for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017

CROSSREFS

Cf. A066325.

Cf. A099174.