Hermite and Laguerre Polynomials

Hermite and Laguerre polynomials are two families of orthogonal polynomials defined on the real line (Hermite, weight

e^{- x^{2}}

) and the half-line (Laguerre, weight

e^{- x}

). They satisfy second-order ODEs, three-term recurrences, and Rodrigues' formulas, and form complete orthogonal bases for their respective weighted

L^{2}

spaces. Where: quantum harmonic oscillator (Hermite), hydrogen radial wavefunctions (Laguerre), Gaussian quadrature, probability (Edgeworth/Gram-Charlier series), numerical analysis. Who/when: named for Charles Hermite (1864) and Edmond Laguerre (1879), though Chebyshev, Laplace, and Abel touched related ideas earlier.

The Hermite polynomials $H_{n} (x)$ and Laguerre polynomials $L_{n} (x)$ are sequences of polynomials, one of each degree $n = 0, 1, 2, \dots$ , that are mutually orthogonal with respect to a fixed weight function. Hermite uses the Gaussian weight $e^{- x^{2}}$ on $(- \infty, \infty)$ ; Laguerre uses $e^{- x}$ on $[0, \infty)$ . They matter because the eigenfunctions of the most important quantum systems—the harmonic oscillator and the hydrogen atom—are built directly from them, and because they provide the optimal node sets for Gauss–Hermite and Gauss–Laguerre numerical integration. A surprising fact: although these polynomials are individually just finite sums of powers, their generating functions are elegant closed-form exponentials, so an entire infinite family is encoded in a single two-variable function. They sit inside the broader theory of Orthogonal Polynomials alongside Legendre and Chebyshev.

The Core Insight

A weight function on an interval defines an inner product $⟨ f, g ⟩ = \int f g w d x$ . Applying Gram–Schmidt to the monomials $1, x, x^{2}, \dots$ under this inner product produces a unique (up to normalization) orthogonal polynomial sequence. The specific choices of weight— $e^{- x^{2}}$ for Hermite, $e^{- x}$ for Laguerre—are exactly the weights that arise from physically natural differential operators, which is why these polynomials appear everywhere quantum mechanics and Gaussian statistics do.

Formal Statement

Hermite Polynomials (physicists' convention)

The Hermite polynomials are defined by Rodrigues' formula

H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}}, n = 0, 1, 2, \dots

⟳repeated differentiation of

e^{- x^{2}}

brings down a polynomial factor times

e^{- x^{2}}

; pulling out

e^{- x^{2}}

and the sign

(- 1)^{n}

isolates that polynomial.

Here $H_{n} (x)$ is a polynomial of degree $n$ with leading coefficient $2^{n}$ . The first few are

H_{0} = 1, H_{1} = 2 x, H_{2} = 4 x^{2} - 2, H_{3} = 8 x^{3} - 12 x, H_{4} = 16 x^{4} - 48 x^{2} + 12.

⟳differentiate

e^{- x^{2}}

once, twice, etc., or use the recurrence [9].

They satisfy the orthogonality relation on $(- \infty, \infty)$ with weight $w (x) = e^{- x^{2}}$ :

\int_{- \infty}^{\infty} H_{m} (x) H_{n} (x) e^{- x^{2}} d x = 2^{n} n! π δ_{mn} .

⟳insert Rodrigues' formula for

H_{m}

, integrate by parts

m

times (boundary terms vanish from the Gaussian), and evaluate the resulting Gaussian integral.

Probabilists' Hermite Polynomials

A second convention, $H e_{n} (x)$ , uses the weight $e^{- x^{2} /2}$ :

H e_{n} (x) = (- 1)^{n} e^{x^{2} /2} \frac{d ^{n}}{d x ^{n}} e^{- x^{2} /2}, H_{n} (x) = 2^{n /2} H e_{n} (2 x) .

⟳substitute

x \mapsto x / 2

into Rodrigues' formula [1] and track the chain-rule factors.

Then $H e_{0} = 1, H e_{1} = x, H e_{2} = x^{2} - 1, H e_{3} = x^{3} - 3 x$ , with leading coefficient $1$ and orthogonality $\int H e_{m} H e_{n} e^{- x^{2} /2} d x = 2 π n! δ_{mn}$ .

Laguerre Polynomials

The Laguerre polynomials are defined by Rodrigues' formula on $[0, \infty)$ :

L_{n} (x) = \frac{e ^{x}}{n !} \frac{d ^{n}}{d x ^{n}} (x^{n} e^{- x}), n = 0, 1, 2, \dots

⟳the Leibniz rule applied to

x^{n} e^{- x}

produces a finite sum; the

e^{x}

cancels the exponential, leaving a degree-

n

polynomial.

Explicitly,

L_{n} (x) = k = 0 \sum n (k n) \frac{( - 1 ) ^{k}}{k !} x^{k},

⟳expand the

n

-th derivative in [5] with the general Leibniz rule and simplify the binomial coefficients.

with $L_{0} = 1, L_{1} = 1 - x, L_{2} = 1 - 2 x + \frac{1}{2} x^{2}, L_{3} = 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3}$ . Orthogonality with weight $w (x) = e^{- x}$ :

\int_{0}^{\infty} L_{m} (x) L_{n} (x) e^{- x} d x = δ_{mn} .

⟳substitute Rodrigues' formula and integrate by parts

n

times; the boundary terms vanish because

x^{n} e^{- x}

and its derivatives die at both endpoints.

Associated (Generalized) Laguerre Polynomials

For a real parameter $α > - 1$ , the associated Laguerre polynomials use weight $x^{α} e^{- x}$ :

L_{n}^{(α)} (x) = \frac{x ^{- α} e ^{x}}{n !} \frac{d ^{n}}{d x ^{n}} (x^{n + α} e^{- x}), \int_{0}^{\infty} L_{m}^{(α)} L_{n}^{(α)} x^{α} e^{- x} d x = \frac{Γ ( n + α + 1 )}{n !} δ_{mn} .

⟳generalize the Laguerre Rodrigues formula by inserting the factor

x^{α}

; the normalization integral is a Gamma function.

Setting $α = 0$ recovers $L_{n}^{(0)} = L_{n}$ . These appear in the hydrogen atom with $α = 2 ℓ + 1$ .

Defining Differential Equations

Each family satisfies a self-adjoint Sturm–Liouville ODE whose polynomial solutions are exactly the $H_{n}$ or $L_{n}$ :

H_{n}^{''} - 2 x H_{n}^{'} + 2 n H_{n} = 0, x L_{n}^{''} + (1 - x) L_{n}^{'} + n L_{n} = 0.

⟳differentiate Rodrigues' formula, or substitute a power-series ansatz and require termination of the series at degree

n

, which forces the eigenvalue

2 n

(resp.

n

For associated Laguerre, $x y^{''} + (α + 1 - x) y^{'} + n y = 0$ . These are the Hermite equation and Laguerre equation; non-negative integer $n$ is precisely the condition for a polynomial (non-singular) solution.

Three-Term Recurrences

Both families obey the universal three-term recurrence of orthogonal polynomials:

H_{n + 1} (x) = 2 x H_{n} (x) - 2 n H_{n - 1} (x),

(n + 1) L_{n + 1} (x) = (2 n + 1 - x) L_{n} (x) - n L_{n - 1} (x) .

⟳orthogonality forces

x p_{n}

to be a combination of

p_{n + 1}, p_{n}, p_{n - 1}

only; matching leading coefficients fixes the constants.

Differentiation relations: $H_{n}^{'} (x) = 2 n H_{n - 1} (x)$ , and $\frac{d}{d x} L_{n}^{(α)} (x) = - L_{n - 1}^{(α + 1)} (x)$ .

Generating Functions

The entire family is packaged in a single exponential:

e^{2 x t - t^{2}} = n = 0 \sum \infty \frac{H _{n} ( x )}{n !} t^{n}, \frac{1}{1 - t} exp (\frac{- x t}{1 - t}) = n = 0 \sum \infty L_{n} (x) t^{n} .

⟳for Hermite, complete the square

- (x - t)^{2} + x^{2}

and Taylor-expand; the coefficients are exactly Rodrigues derivatives. For Laguerre, expand the exponential and resum.

Why It Holds / Derivation

Orthogonality of Hermite Polynomials via Rodrigues

We prove [3] for $m < n$ (the $m = n$ case gives the norm).

Setup. Write $I = \int_{- \infty}^{\infty} H_{m} (x) H_{n} (x) e^{- x^{2}} d x$ and substitute Rodrigues' formula $H_{n} e^{- x^{2}} = (- 1)^{n} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}}$ :

I = (- 1)^{n} \int_{- \infty}^{\infty} H_{m} (x) \frac{d ^{n}}{d x ^{n}} e^{- x^{2}} d x .

Integrate by parts $n$ times. Each integration moves one derivative from $e^{- x^{2}}$ onto $H_{m}$ . Boundary terms are products of polynomials with $e^{- x^{2}} \to 0$ , hence vanish. After $n$ steps:

I = (- 1)^{n} (- 1)^{n} \int_{- \infty}^{\infty} \frac{d ^{n} H _{m}}{d x ^{n}} e^{- x^{2}} d x = \int_{- \infty}^{\infty} H_{m}^{(n)} (x) e^{- x^{2}} d x .

Degree argument. Since $de g H_{m} = m < n$ , the $n$ -th derivative $H_{m}^{(n)} \equiv 0$ . Therefore $I = 0$ , proving orthogonality.

Norm ( $m = n$ ). Then $H_{n}^{(n)} = 2^{n} n!$ (the leading coefficient is $2^{n}$ , and the $n$ -th derivative of $x^{n}$ is $n!$ ). So

I = 2^{n} n! \int_{- \infty}^{\infty} e^{- x^{2}} d x = 2^{n} n! π . ■

Hermite ODE from the Generating Function

Let $G (x, t) = e^{2 x t - t^{2}} = \sum_{n} \frac{H _{n}}{n !} t^{n}$ .

Compute $\partial_{x} G = 2 t G$ . Matching $t^{n}$ coefficients gives $H_{n}^{'} = 2 n H_{n - 1}$ (the differentiation relation).
Compute $\partial_{t} G = (2 x - 2 t) G$ . Matching coefficients gives $H_{n + 1} = 2 x H_{n} - 2 n H_{n - 1}$ (recurrence [10]).
Differentiate the recurrence: $H_{n + 1}^{'} = 2 H_{n} + 2 x H_{n}^{'} - 2 n H_{n - 1}^{'}$ . Use $H_{n}^{'} = 2 n H_{n - 1}$ throughout to express everything in $H_{n}, H_{n}^{'}$ .
Substituting $H_{n + 1}^{'} = 2 (n + 1) H_{n}$ and $H_{n - 1}^{'} = H_{n}^{'} / (2 n)$ and simplifying yields $H_{n}^{''} - 2 x H_{n}^{'} + 2 n H_{n} = 0$ , the Hermite equation [9]. $■$

Charles Hermite | 1822–1901

Hermite was a French mathematician of immense influence whose name attaches to Hermitian matrices, Hermite interpolation, Hermite normal form, and these polynomials. In 1864 he studied the polynomials systematically in connection with the heat equation and probability. Famously, in 1873 he proved that $e$ is a transcendental number—a result later extended by Lindemann to $π$ . Despite a lifelong disability in one leg and chronic ill health, Hermite mentored a generation including Poincaré and Stieltjes, and championed the analytic study of special functions. He distrusted "monsters" like nowhere-differentiable functions, once writing that he turned away "with fright and horror" from them.

Building Intuition

Think of the weight function as a spotlight. The Gaussian $e^{- x^{2}}$ shines brightest at the origin and fades symmetrically; orthogonality "under that spotlight" means $H_{m}$ and $H_{n}$ have no overlap when weighted by it. Each successive polynomial $H_{n}$ must wiggle so as to be perpendicular to all lower ones—and the only way a polynomial of degree $n$ can do that is to acquire exactly $n$ real zeros, all inside the region where the weight is significant. The zeros spread out roughly like $n$ for Hermite (matching the classical turning points of an oscillator with energy $n$ ) and crowd near $0$ for Laguerre. The generating function is the "DNA": a single Gaussian-in- $t$ (Hermite) or rational-exponential (Laguerre) whose Taylor coefficients are the whole family, reflecting that these polynomials are the natural eigenfunctions of translation-and-scaling operators built from $\frac{d}{d x}$ and multiplication by $x$ .

Hermite polynomials $H_0$–$H_3$ over the Gaussian weight region

Worked Examples

Gauss–Hermite Quadrature with 2 nodes

Given: Approximate

I = \int_{- \infty}^{\infty} e^{- x^{2}} x^{4} d x

using 2-point Gauss–Hermite quadrature, then compare to the exact value. Steps.

The nodes of $n$ -point Gauss–Hermite are the roots of $H_{n}$ . For $H_{2} = 4 x^{2} - 2 = 0$ , roots are $x = \pm \frac{1}{2}$ .
The weights are $w_{i} = \frac{2 ^{n - 1} n ! π}{n ^{2} [ H _{n - 1} ( x _{i} ) ] ^{2}}$ . With $n = 2$ , $H_{1} (x_{i}) = 2 x_{i} = \pm 2$ , so $[H_{1}]^{2} = 2$ :

w_{i} = \frac{2 ^{1} \cdot 2 ! \cdot π}{4 \cdot 2} = \frac{4 π}{8} = \frac{π}{2} .

Quadrature estimate: $I \approx \sum w_{i} f (x_{i}) = \frac{π}{2} [(\frac{1}{2})^{4} + (- \frac{1}{2})^{4}] = \frac{π}{2} \cdot 2 \cdot \frac{1}{4} = \frac{π}{4} .$
Exact value: $\int_{- \infty}^{\infty} x^{4} e^{- x^{2}} d x = \frac{3}{4} π$ .

Answer. The 2-node rule gives

\frac{π}{4} \approx 0.443

versus exact

\frac{3}{4} π \approx 1.329

. The estimate is wrong because 2-point Gauss–Hermite is exact only up to degree

2 n - 1 = 3

;

x^{4}

exceeds that. (A 3-point rule would be exact.) This illustrates the degree-

2 n - 1

guarantee.

Hydrogen-like radial normalization with associated Laguerre

Given: Verify the orthonormality scale for the

n = 2, ℓ = 0

hydrogen radial part, which is proportional to

L_{1}^{(1)} (ρ)

with

ρ = r / a_{0}

(in suitable units). Compute

\int_{0}^{\infty} [L_{1}^{(1)} (x)]^{2} x^{1} e^{- x} d x

. Steps.

From [6]–[8], $L_{1}^{(1)} (x) = 2 - x$ (since $L_{n}^{(α)}$ with $n = 1$ is $(α + 1) - x$ , here $α = 1$ ).
Use the normalization integral from [8]: $\int_{0}^{\infty} [L_{n}^{(α)}]^{2} x^{α} e^{- x} d x = \frac{Γ ( n + α + 1 )}{n !}$ .
Plug $n = 1, α = 1$ : $\frac{Γ ( 3 )}{1 !} = \frac{2 !}{1} = 2.$
Direct check: $\int_{0}^{\infty} (2 - x)^{2} x e^{- x} d x = \int_{0}^{\infty} (4 x - 4 x^{2} + x^{3}) e^{- x} d x = 4 (1!) - 4 (2!) + 3! = 4 - 8 + 6 = 2.$ ✓

Answer.

\int_{0}^{\infty} [L_{1}^{(1)} (x)]^{2} x e^{- x} d x = 2

, confirming the Gamma-function normalization that fixes the hydrogen wavefunction constant.

Common Misconceptions

The two Hermite conventions are not interchangeable. Physicists'

H_{n}

(weight

e^{- x^{2}}

, leading coeff

2^{n}

) and probabilists'

H e_{n}

(weight

e^{- x^{2} /2}

, leading coeff

1

) differ by the scaling [4]. A formula copied from a statistics text into a quantum-mechanics calculation will be wrong by factors of

2

and

2^{n /2}

. Always check the weight first. Likewise, some references define Laguerre with an extra

n!

(so that the constant term is

n!

instead of

1

)—verify

L_{0} = 1

and the normalization [7] before trusting tables. "Any polynomial of degree $n$ orthogonal to lower degrees is Hermite." Orthogonality alone does not fix a family—it depends entirely on the weight. The polynomial

x^{2} - \frac{1}{3}

is orthogonal to

1

and

x

[- 1, 1]

with weight

1

(it is the Legendre

P_{2}

up to scale), but

H_{2} = 4 x^{2} - 2

corresponds to weight

e^{- x^{2}}

. They are different polynomials. Orthogonal-polynomial identity is determined by the pair (interval, weight), not by orthogonality in the abstract.

Connections

Hermite and Laguerre are two members of the classical orthogonal polynomials, the others being Legendre Polynomials (weight $1$ on $[- 1, 1]$ ) and Chebyshev (weight $(1 - x^{2})^{- 1/2}$ ). All four arise as polynomial eigenfunctions of Sturm-Liouville Theory|Sturm–Liouville problems and share the structural triad: Rodrigues formula, three-term recurrence, generating function. Hermite functions $ψ_{n} (x) = H_{n} (x) e^{- x^{2} /2}$ are the eigenfunctions of the Quantum Harmonic Oscillator and, remarkably, also the eigenfunctions of the Fourier Transform (with eigenvalues $(- i)^{n}$ ). The Laguerre family enters the radial Schrödinger Equation for the hydrogen atom. Both are special cases of the hypergeometric/confluent-hypergeometric functions, linking them to the broader Special Functions landscape.

Real-World Applications

Quantum harmonic oscillator. The stationary states are $ψ_{n} (x) = (\frac{mω}{π ℏ})^{1/4} \frac{1}{2 ^{n} n !} H_{n} (ξ) e^{- ξ^{2} /2}$ , $ξ = mω /ℏ x$ . Used in molecular vibration spectroscopy and quantum optics (photon number states).
Hydrogen atom. Radial wavefunctions $R_{n ℓ} (r) \propto ρ^{ℓ} e^{- ρ /2} L_{n - ℓ - 1}^{(2 ℓ + 1)} (ρ)$ , foundational to all of atomic physics and chemistry.
Gauss–Hermite & Gauss–Laguerre quadrature. Optimal numerical integration of $\int e^{- x^{2}} f$ and $\int_{0}^{\infty} e^{- x} f$ , used in computational physics, statistics, and Bayesian inference (marginalizing Gaussian integrals).
Gram–Charlier and Edgeworth expansions. Probabilists' Hermite polynomials expand a probability density as corrections to a Gaussian, used in finance (option pricing with skew/kurtosis) and statistical mechanics.
Laser beam modes. Hermite–Gaussian ( $TEM_{mn}$ ) modes describe the transverse field of a laser cavity with rectangular symmetry; Laguerre–Gaussian modes carry orbital angular momentum and are used in optical tweezers and quantum communication.
Signal processing. Hermite functions form a basis well-localized in both time and frequency, used in feature extraction for ECG/EEG analysis.
Density functional & quantum chemistry codes. Laguerre-type orbitals and Gauss–Hermite grids underpin numerical evaluation of molecular integrals.

Historical Development

1810 — Pierre-Simon Laplace encounters Hermite-type polynomials in his work on probability and the central limit theorem.
1826 — Niels Henrik Abel and others touch on related polynomial sequences in studies of integrals.
1859 — Pafnuty Chebyshev develops orthogonal polynomials systematically, including the weight $e^{- x^{2}}$ case, predating Hermite's paper.
1864 — Charles Hermite publishes Sur un nouveau développement en série des fonctions, introducing what are now called Hermite polynomials in connection with the heat equation.
1879 — Edmond Laguerre studies the polynomials bearing his name while investigating the integral $\int_{x}^{\infty} e^{- t} t^{- 1} d t$ (related to the exponential integral).
1880s — Thomas Stieltjes and others develop the general theory of orthogonal polynomials and continued fractions, placing Hermite/Laguerre in a unified framework.
1926 — Erwin Schrödinger solves the quantum harmonic oscillator and hydrogen atom, revealing Hermite and Laguerre polynomials as the natural eigenfunctions—making them central to physics.
1929 — Gábor Szegő's treatise Orthogonal Polynomials (published 1939) consolidates the asymptotic and analytic theory.
1966 — Hermite–Gaussian laser modes are identified experimentally as the natural transverse modes of stable optical resonators.

Subtleties & Limits

The fine structure of the zeros and asymptotics of these polynomials remains an active research area. The Plancherel–Rotach asymptotics describe $H_{n}$ in the oscillatory (bulk) region versus the exponential (turning-point) region, and the precise behavior near the edge is governed by Airy functions—a connection to random matrix theory, where the largest eigenvalue of a Gaussian ensemble obeys the Tracy–Widom distribution built from Hermite asymptotics. Open and subtle questions include: sharp bounds on the smallest gap between consecutive zeros, the behavior of generalized Laguerre polynomials $L_{n}^{(α)}$ when $α$ grows with $n$ (relevant to high-dimensional statistics), and the analytic theory of exceptional orthogonal polynomials (discovered ~2009), which are Hermite/Laguerre-like families missing certain degrees yet still complete and solvable—expanding the catalog of exactly solvable quantum potentials beyond the classical ones.