mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Hermite numbers are values of

Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...

at zero argument. Typically they are defined for physicists' Hermite polynomials.

Formal definition

The numbers ''H''_n = ''H''_n(0), where ''H''_n(''x'') is a

Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as i ...

of order ''n'', may be called Hermite numbers.Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html The first Hermite numbers are: :

H_0 = 1\,

H_1 = 0\,

H_2 = -2\,

H_3 = 0\,

H_4 = +12\,

H_5 = 0\,

H_6 = -120\,

H_7 = 0\,

H_8 = +1680\,

H_9 =0\,

H_ = -30240\,

Recursion relations

Are obtained from

recursion relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s of Hermitian polynomials for ''x'' = 0: :

H_ = -2(n-1)H_.\,\!

Since ''H''₀ = 1 and ''H''₁ = 0 one can construct a closed formula for ''H''_n: :

H_n = 
\begin 
  0,  & \mboxn\mbox \\
  (-1)^ 2^ (n-1)!! , & \mboxn\mbox 
\end

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

Usage

From the

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...

of Hermitian polynomials it follows that :

\exp (-t^2 + 2tx) = \sum_^\infty H_n (x) \frac {n!}\,\!

Reference gives a

formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...

: :

H_n (x) = (H+2x)^n\,\!

where formally the ''n''-th power of ''H'', ''H''^''n'', is the ''n''-th Hermite number, ''H''_''n''. (See

Umbral calculus In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...

Notes

Integer sequences