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# Hermite number

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

Formal definition

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

The first Hermite numbers are:

\( H_0 = 1\, \)

\( H_1 = 0\, \)

\( H_2 = -2\, \)

\( H_3 = 0\, \)

\( H_4 = +12\, \)

\( H_5 = 0\, \)

\( H_6 = -120\, \)

\( H_7 = 0\, \)

\( H_8 = +1680\, \)

\( H_9 =0\, \)

\( H_{10} = -30240\, \)

Recursion relations

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

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

Since H_{0} = 1 and H_{1} = 0 one can construct a closed formula for Hn:

\( H_n = \begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} \)

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

Usage

From the generating function of Hermitian polynomials it follows that

\( \exp (t^2) = \sum_{n=0}^\infty H_n \frac {t^n}{n!}\,\! \)

Reference [1] gives a formal power series:

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

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

Notes

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

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