HermiteH

The physicists' Hermite polynomial $H_n(x)$, orthogonal on the real line with weight $e^{-x^2}$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHermite polynomials↗MathWorldHermitePolynomial↗DLMF18.3.T1.t1.r14↗Wolfram LanguageHermiteH✓↗

HermiteH(n, x)the degree-$n$ Hermite polynomial at $x$, for an integer $n \ge 0$.

Typed signatures

Type	Package	Applies
(number, number) -> number	analytic	always

Domain: Special functions

Details

• $H_0 = 1$, $H_1 = 2x$, and $H_{n+1} = 2x\,H_n - 2n\,H_{n-1}$. The coefficients are integers and the leading one is $2^n$.
• $H_n' = 2n\,H_{n-1}$, so D(HermiteH(n, x), x) is $2n\,H_{n-1}(x)$ at a symbolic order too.
• $\int_{-\infty}^{\infty} H_m(x)\,H_n(x)\,e^{-x^2}\,dx = 2^n\,n!\,\sqrt\pi\,\delta_{mn}$.
• Parity follows the degree: $H_n(-x) = (-1)^n H_n(x)$, and $H_{2k}(0) = (-1)^k (2k)!/k!$.
• At an integer $n$ and a symbolic or exact $x$ the result is the expanded polynomial. At a floating-point or complex $x$ the recurrence runs directly on the number, and the head stays unevaluated if it overflows the double range.
• A symbolic order stays unevaluated. Wolfram continues $H_\nu$ to negative and non-integer $\nu$ through confluent hypergeometric functions, and that is not implemented here; those orders also stay unevaluated.

Examples

3 more kept as data, checked against the oracles

Basic #

$H_2(x) = 4x^2 - 2$

live ↗

In$\mathrm{HermiteH}(2, x)$Out$4x^2-2$

$H_3(x) = 8x^3 - 12x$

live ↗

In$\mathrm{HermiteH}(3, x)$Out$8x^3-12x$

At an integer the value is exact: $16 \cdot 16 - 48 \cdot 4 + 12$

live ↗

In$\mathrm{HermiteH}(4, 2)$Out$76$

At a floating-point $x$

live ↗

In$\mathrm{HermiteH}(5, 0.7)$Out$34.498\,24$

$H_n'(x) = 2n\,H_{n-1}(x)$

live ↗

In$\frac{\mathrm{d}\mathrm{HermiteH}(n, x)}{\mathrm{d}x}$Out$2n\mathrm{HermiteH}(n-1, x)$

Scope #

At a rational $x$ the value is an exact rational

live ↗

In$\mathrm{HermiteH}(5, \frac{1}{3})$Out$\frac{8\,312}{243}$

At a Gaussian integer

live ↗

In$\mathrm{HermiteH}(4, 2+3i)$Out$-1\,652-2\,496i$

A symbolic order stays unevaluated

live ↗

In$\mathrm{HermiteH}(n, x)$Out$\mathrm{HermiteH}(n, x)$

The degree-10 polynomial in $x$

live ↗

In$\mathrm{HermiteH}(10, x)$Out$1\,024x^{10}-23\,040x^8+161\,280x^6-403\,200x^4+302\,400x^2-30\,240$

Properties #

$H_{2k}(0) = (-1)^k (2k)!/k!$

live ↗

In$\mathrm{HermiteH}(6, 0)$Out$-120$

Odd degree, odd function: $H_n(-x) = (-1)^n H_n(x)$

live ↗

In$\mathrm{HermiteH}(5, -\frac{1}{3})=-\mathrm{HermiteH}(5, \frac{1}{3})$Out$\mathrm{True}$

$H_{n+1} = 2x\,H_n - 2n\,H_{n-1}$, here at $n = 4$, $x = 3$

live ↗

In$\mathrm{HermiteH}(5, 3)-(6\mathrm{HermiteH}(4, 3)-8\mathrm{HermiteH}(3, 3))$Out$0$

About

Examples	15
Checked against	Wolfram Language 15 agree
Crosswalk	WikipediaMathWorldDLMFWolfram Language

Implementation

referenceenginenotatio

The explicit sum over $m$ up to $\lfloor n/2 \rfloor$; the kernel runs the three-term recurrence instead, and the two are checked against each other.

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hermite.ts

mappedexternalwolfram / mpmath / sympypackages/oracle/src/mappings.ts

HermiteH[n, x]; mpmath.hermite(n, x); sympy.hermite(n, x).

See also: ChebyshevT, LegendrePolynomial, LaplaceTransform