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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.

Contents

Formal definition

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

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

H n = { 0 , if  n  is odd ( − 1 ) n / 2 2 n / 2 ( n − 1 ) ! ! , if  n  is even

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

Usage

From the generating function of Hermitian polynomials it follows that

exp ⁡ ( − t 2 ) = ∑ n = 0 ∞ H n t n n !

Reference gives a formal power series:

H n ( x ) = ( H + 2 x ) n

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

References

Hermite number Wikipedia
(Text) CC BY-SA


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