Hermite numbers

The Hermite numbers Hn  may be defined by the generating function

e-t2:=n=0Hntnn! (1)

which is same as the generating function of Hermite polynomials at the value 0 of the argumentMathworldPlanetmath z.  After expanding the left hand side of (1) to Taylor seriesMathworldPlanetmath 1-t21!+t42!-t63!+-, one can write

1-2t22!+12t44!-120t66!+-=n=0Hntnn!. (2)

Thus one sees that



Hn={(-1)n2n!(n2)! when 2n0    when 2n (3)

The Hermite numbers form the sequence (http://www.research.att.com/ njas/sequences/index.html?q=A067994&language=english&go=SearchSloane A067994)

1, 0,-2, 0, 12, 0,-120, 0, 1680, 0,-30240, 0, 665280, 0,-17297280, 0,

which obeys the recurrence relation

Hn= 2(1-n)Hn-2. (4)

According to (1), the Hermite numbers satisfy  Hn=Hn(0)  where Hn(x) is the Hermite polynomialDlmfDlmfDlmfMathworldPlanetmath of degree n.  The of Hermite numbers and Hermite polynomials may be expressed also by using symbolic powers


as follows:

(2x+H)n=Hn(x). (5)

This means e.g. that

(2x+H)2=(2x)2+22xH1+H2= 4x2+4xH1+H2= 4x2-2=H2(x).
Title Hermite numbers
Canonical name HermiteNumbers
Date of creation 2013-03-22 19:08:32
Last modified on 2013-03-22 19:08:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 11B68
Related topic EulerNumbers2
Related topic AppellSequence