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Hermite Number


The numbers H_n=H_n(0), where H_n(x) is a Hermite polynomial, may be called Hermite numbers. For n=0, 1, ..., the first few are 1, 0, -2, 0, 12, 0, -120, 0, 1680, 0, ... (OEIS A067994). They are given explicitly by

H_n=(2^nsqrt(pi))/(Gamma(1/2(1-n)))
(1)
={0 for odd n; ((-1)^(n/2)n!)/((1/2n)!) for even n.
(2)

As a result of the ratio n!/(n/2)! always being divisible by n for n>2, the only prime Hermite number is H_2=-2.

The Hermite numbers H_n are related to the Hermite polynomials H_n(x) by

 H_n(x)=(H+2x)^n,
(3)

where H^k=H_k, and

 H_n=(H-2x)^n,
(4)

where H^k=H_k(x).


See also

Hermite Polynomial

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References

Sloane, N. J. A. Sequence A067994 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Hermite Number

Cite this as:

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

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