Hermite Number

Author

Eric W. Weisstein
September 14, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/HermiteNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HermiteNumber.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

Unprotect[HermiteH];
HermiteH[n_]:=HermiteH[n,0]
Protect[HermiteH];

Enumeration

HermiteH[n,0]

FullSimplify[(((I^n)n!)/((n/2)!))-(((2^n) (Sqrt[π]))/(Gamma[((1-n)/2)])),Mod[n,2]==0&&n>=0]

Table[(((I^n)n!)/((n/2)!))-(((2^n) (Sqrt[π]))/(Gamma[((1-n)/2)])),{n,0,10,2}]

Table[HermiteH[n],{n,0,10}]

Table[(((2^n) (Sqrt[π]))/(Gamma[(1/2)-(n/2)])),{n,0,10}]

Table[If[OddQ[n],0,(((((-1))^(n/2))n!)/(((1/2)n)!))],{n,0,10}]

Generating Function

GF

Sum[HermiteH[n,0](x^n),{n,0,∞}]

Sum[HermiteH[n,0](x^n),{n,0,10}]

GeneratingFunction[HermiteH[n,0],n,x]

EGF

Sum[HermiteH[n,0]((x^n)/(n!)),{n,0,∞}]//InputForm

Sum[HermiteH[n,0]((x^n)/(n!)),{n,0,10}]

ExponentialGeneratingFunction[HermiteH[n,0],n,x]

SeriesCoefficient[Exp[-t^2],{t,0,n}]//InputForm

Sum[x^n*Piecewise[{{I^n/(n/2)!, Mod[n, 2] == 0 && n >= 0}}, 0], {n, 0, Infinity}]//InputForm

Sum[((I^n)/((n/2)!))(x^n),{n,0,∞,2}]

Sum[((((-1))^n)/(n!))(x^(2n)),{n,0,∞}]

Primes

FullSimplify[(((((-1))^(n/2))n!)/(((1/2)n)!))/.n->2k,k∈Integers]

Position[Table[(2k)!/k!/(2k),{k,2,100}],_Rational]

So can never be prime except 2

Position[Table[(((((-1))^k) (2 k)!)/(k!)),{k,0,10^3}],_?PrimeQ]//Timing

Identities

(Subscript[H, n])(x)==(((H+2 x))^n)

Subtract@@@Expand[Table[{HermiteH[n,x],Expand[(H+2x)^n]},{n,0,10}]/.H^n_.:>HermiteH[n,0]]//Union

(Subscript[H, n])==(((H-2 x))^n)

Subtract@@@Expand[Table[{HermiteH[n,0],Expand[(H-2x)^n]},{n,0,10}]/.H^n_.:>HermiteH[n,x]]//Union