Hermite Polynomial Hₙ(x) Calculator

Evaluate the physicists Hermite polynomial Hₙ(x) by a three term recurrence. See neighbouring degrees, the derivative, the probabilists Heₙ(x), a coefficient table and a graph.

Input

Compute the physicists Hermite polynomial Hₙ(x) using a three term recurrence. Enter a degree n and a value x.

Degree n

A non negative integer

Value of x

Any real number

Result

Value of Hₙ(x) with n = 4

-12.1664

at x = 0.8

Degree n

H(n-1)(x)

-5.504

H(n+1)(x)

24.56576

Derivative Hn'(x)

-44.032

Probabilists Heₙ(x)

-0.4304

Graph of Hₙ(x) with n = 4

Shape for x from -3 to 3. The entered x is marked with a dot.

Coefficients of Hₙ(x) with n = 4

Coefficients for each power of x, listed from the highest. Terms with a zero coefficient are omitted.

Power of x	Coefficient
4	16
2	-48
0	12

How it works

The physicists Hermite polynomial uses H0(x)=1, H1(x)=2x and the three term recurrence H(n+1)=2x Hn(x)-2n H(n-1)(x).
The derivative is obtained from the relation Hn'(x)=2n H(n-1)(x).
The probabilists version Heₙ(x) uses He0(x)=1, He1(x)=x and He(n+1)=x Hen(x)-n He(n-1)(x). The two are related by H_n(x)=2^(n/2) He_n(x times sqrt(2)).
For large degrees the values grow very fast, so floating point overflow and rounding error appear. The degree is capped at 200 as a practical limit.
The graph shows the shape of Hₙ(x) for x from -3 to 3 and marks the entered x with a dot.

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