Hermite number

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In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.

Formal definition[edit]

The numbers Hn = Hn(0), where Hn(x) is a Hermite polynomial of order n, may be called Hermite numbers.[1]

The first Hermite numbers are:

Recursion relations[edit]

Are obtained from recursion relations of Hermitian polynomials for x = 0:

Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn:

where (n - 1)!! = 1 × 3 × ... × (n - 1).


From the generating function of Hermitian polynomials it follows that

Reference [1] gives a formal power series:

where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. (See Umbral calculus.)


  1. ^ a b Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html