Laguerre and Associated Laguerre Polynomials

Laguerre polynomials

L_{n} (x)

are a sequence of orthogonal polynomials on

[0, \infty)

with respect to the weight

e^{- x}

. The associated (generalized) Laguerre polynomials

L_{n}^{(α)} (x)

generalize these with weight

x^{α} e^{- x}

. They solve Laguerre's differential equation, satisfy a three-term recurrence, and are central to the radial wavefunctions of the hydrogen atom. Introduced by Edmond Laguerre (1879) and developed through the 19th–20th centuries.

The Laguerre polynomials form a complete orthogonal basis of polynomials on the half-line $[0, \infty)$ under the weight function $e^{- x}$ . They arise naturally whenever one solves a second-order linear ODE of "confluent hypergeometric" type, most famously the radial Schrödinger equation for the hydrogen atom and quantum harmonic-oscillator-related problems. The generalized version $L_{n}^{(α)}$ introduces a parameter $α > - 1$ that tilts the weight toward $x^{α} e^{- x}$ , broadening their applicability to Gauss–Laguerre quadrature and Coulomb problems. A surprising fact: although defined on an infinite interval, their orthogonality, recurrence, and generating function are as clean as those of the classical polynomials on bounded intervals. They are one of the three "classical orthogonal polynomial" families (alongside Hermite and Jacobi).

The Core Insight

Demanding that a power-series solution of a confluent-type ODE terminate (so the wavefunction stays normalizable) forces an integer eigenvalue $n$ . The resulting finite polynomials are automatically orthogonal under $x^{α} e^{- x}$ because they are eigenfunctions of a self-adjoint (Sturm–Liouville) operator. Orthogonality, recurrence, and the Rodrigues formula are all manifestations of this single structure.

Formal Statement

Laguerre Polynomials

The Laguerre polynomials $L_{n} (x)$ , for $n = 0, 1, 2, \dots$ , are defined by the Rodrigues formula

L_{n} (x) = \frac{e ^{x}}{n !} \frac{d ^{n}}{d x ^{n}} (e^{- x} x^{n}) .

⟳Apply Leibniz's rule for the

n

-th derivative of the product

e^{- x} x^{n}

to extract the explicit polynomial.

Equivalently, they have the explicit closed form

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

⟳Expand the Rodrigues formula via Leibniz's rule and use

\frac{d ^{n}}{d x ^{n}} (x^{n + k - ...})

; collect terms.

where $x \in [0, \infty)$ , $n \in Z_{\geq 0}$ , $(k n)$ is the binomial coefficient, and $L_{0} (x) = 1$ .

Associated (Generalized) Laguerre Polynomials

For a real parameter $α > - 1$ , the associated Laguerre polynomials are

L_{n}^{(α)} (x) = \frac{x ^{- α} e ^{x}}{n !} \frac{d ^{n}}{d x ^{n}} (e^{- x} x^{n + α}) = k = 0 \sum n (- 1)^{k} (n - k n + α) \frac{x ^{k}}{k !} .

⟳Generalize the Rodrigues formula by inserting

x^{α}

into the weight; expand with Leibniz's rule, identifying

(n - k n + α) = \frac{Γ ( n + α + 1 )}{Γ ( k + α + 1 ) ( n - k )!}

When $α = 0$ these reduce to the ordinary Laguerre polynomials: $L_{n}^{(0)} (x) = L_{n} (x)$ . The symbol $Γ$ denotes the Gamma function, extending the factorial to non-integer arguments.

Laguerre Differential Equation

L_{n}^{(α)} (x)

is the polynomial solution of the second-order linear ODE

x y^{''} + (α + 1 - x) y^{'} + n y = 0,

⟳Substitute the series for

L_{n}^{(α)}

and verify term by term, or recognize it as the eigenvalue equation of the Laguerre operator with eigenvalue

n

with eigenvalue $n$ . For $α = 0$ this is Laguerre's equation $x y^{''} + (1 - x) y^{'} + n y = 0$ . The polynomial solution exists (and is normalizable under the weight) only for non-negative integer $n$ .

Orthogonality Relation

Over $[0, \infty)$ with weight $w (x) = x^{α} e^{- x}$ ,

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

⟳Cast equation [4] into Sturm–Liouville form

\frac{d}{d x} (x^{α + 1} e^{- x} y^{'}) = - n x^{α} e^{- x} y

; self-adjointness gives

δ_{mn}

, and the norm follows from the Rodrigues formula via repeated integration by parts.

where $δ_{mn}$ is the Kronecker delta. For $α = 0$ the norm is exactly $1$ , so the $L_{n} (x)$ are orthonormal under $e^{- x}$ .

Three-Term Recurrence and Generating Function

The polynomials satisfy the recurrence

(n + 1) L_{n + 1}^{(α)} (x) = (2 n + α + 1 - x) L_{n}^{(α)} (x) - (n + α) L_{n - 1}^{(α)} (x),

⟳Compare coefficients of

t^{n}

in the derivative of the generating function [7] with respect to

t

, or apply Favard's theorem for orthogonal polynomials.

and possess the generating function

n = 0 \sum \infty L_{n}^{(α)} (x) t^{n} = \frac{1}{( 1 - t ) ^{α + 1}} exp (\frac{- x t}{1 - t}), ∣ t ∣ < 1.

⟳Expand the exponential and the binomial

(1 - t)^{- (α + 1)}

as power series in

t

, collect the coefficient of

t^{n}

, and match equation [3].

Why It Holds / Derivation

Orthogonality from Sturm–Liouville structure

Step 1. Rewrite Laguerre's equation [4] in self-adjoint (Sturm–Liouville) form. Multiply by the integrating factor

x^{α} e^{- x}

\frac{d}{d x} (x^{α + 1} e^{- x} y^{'}) = - n x^{α} e^{- x} y .

⟳Note

\frac{d}{d x} (x^{α + 1} e^{- x}) = x^{α} e^{- x} (α + 1 - x)

, which reproduces the first-order coefficient in [4].

Step 2. Let

L_{m}^{(α)}

L_{n}^{(α)}

correspond to eigenvalues

m, n

. Write [8] for each, multiply the

m

-equation by

L_{n}^{(α)}

and the

n

-equation by

L_{m}^{(α)}

, subtract, and integrate over

[0, \infty)

. Step 3. The left side becomes a total derivative of

x^{α + 1} e^{- x} (L_{m}^{'} L_{n} - L_{n}^{'} L_{m})

, which vanishes at

x = 0

(the factor

x^{α + 1}

kills it for

α > - 1

) and at

x = \infty

(the factor

e^{- x}

dominates). Hence the boundary term is zero. Step 4. What remains is

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

. For

m \neq = n

this forces the integral to vanish, giving orthogonality. Step 5. For the norm (

m = n

), substitute the Rodrigues formula [3] and integrate by parts

n

times. Each integration moves a derivative off

e^{- x} x^{n + α}

; boundary terms vanish, and the surviving integral is

\frac{1}{n !} \int_{0}^{\infty} x^{n + α} e^{- x} d x \cdot (leading coefficient factor)

. Evaluating with

\int_{0}^{\infty} x^{n + α} e^{- x} d x = Γ (n + α + 1)

yields the norm

Γ (n + α + 1) / n!

in [5].

■

Edmond Nicolas Laguerre | 1834–1886

A French mathematician and artillery officer, Laguerre worked at the École Polytechnique and the Bureau des Longitudes. In 1879 he introduced the polynomials now bearing his name while studying the integral $\int_{x}^{\infty} \frac{e ^{- t}}{t} d t$ and continued-fraction expansions. His broader contributions span geometry (Laguerre transformations, the Laguerre formula for angles), the theory of equations, and approximation theory. The polynomials were a side product of analytic work but became indispensable a half-century later when quantum mechanics revealed them as the radial structure of the hydrogen atom.

Building Intuition

Think of the weight $e^{- x}$ as defining a "probability cloud" concentrated near the origin and decaying on the half-line. The Laguerre polynomials are the unique polynomials that are mutually perpendicular when measured against this cloud — they are the half-line analogue of cosines on a circle. Each $L_{n}$ has exactly $n$ real positive roots, and as $n$ grows these roots spread out to sample more of the half-line, much like higher harmonics oscillate faster. The parameter $α$ controls how strongly the weight is pushed away from $x = 0$ : larger $α$ means the cloud thins out at the origin (the $x^{α}$ factor suppresses small $x$ ), shifting where the polynomials "live."

:::svg[The first few Laguerre polynomials $L_{0}$ through $L_{3}$ on $[0, 5]$ , showing increasing oscillation and roots]

Worked Examples

Computing $L_3(x)$ and verifying a root

Given: Use the explicit formula [2] for

n = 3

α = 0

. Step 1. Apply

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

with

n = 3

$k = 0$ : $(0 3) \frac{1}{0 !} = 1$
$k = 1$ : $(1 3) \frac{- 1}{1 !} = - 3$
$k = 2$ : $(2 3) \frac{1}{2 !} = 3 \cdot \frac{1}{2} = \frac{3}{2}$
$k = 3$ : $(3 3) \frac{- 1}{3 !} = - \frac{1}{6}$

Step 2. Assemble:

L_{3} (x) = 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} .

⟳Sum the four computed coefficients against powers of

x

Step 3. Verify the recurrence-free value at

x = 0

L_{3} (0) = 1

, consistent with

L_{n} (0) = 1

for all

n

(set

x = 0

in [2], only

k = 0

survives). Step 4. Check it solves [4] at, say, evaluating the ODE residual symbolically:

x y^{''} + (1 - x) y^{'} + 3 y

with

y = L_{3}

gives

0

identically (one may verify the cubic coefficient:

y^{''} = 3 - x

y^{'} = - 3 + 3 x - \frac{1}{2} x^{2}

L_{3} (x) = 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} (dimensionless)

Gauss–Laguerre quadrature of an integral

Given: Approximate

I = \int_{0}^{\infty} e^{- x} x cos x d x

using 2-point Gauss–Laguerre quadrature. The nodes are the roots of

L_{2} (x) = 1 - 2 x + \frac{1}{2} x^{2}

, and the weights are

w_{i} = \frac{x _{i}}{( n + 1 ) ^{2} [ L _{n + 1} ( x _{i} ) ] ^{2}}

. Step 1. Roots of

L_{2}

: solve

\frac{1}{2} x^{2} - 2 x + 1 = 0 \Rightarrow x^{2} - 4 x + 2 = 0 \Rightarrow x = 2 \pm 2

. So

x_{1} = 0.585786

x_{2} = 3.414214

. Step 2. Weights (standard tabulated values for

n = 2

w_{1} = 0.853553

w_{2} = 0.146447

. Step 3. The rule approximates

\int_{0}^{\infty} e^{- x} f (x) d x \approx \sum_{i} w_{i} f (x_{i})

with

f (x) = x cos x

$f (x_{1}) = 0.585786 cos (0.585786) = 0.585786 \times 0.83366 = 0.488352$
$f (x_{2}) = 3.414214 cos (3.414214) = 3.414214 \times (- 0.96402) = - 3.29154$

Step 4. Sum:

I \approx w_{1} f (x_{1}) + w_{2} f (x_{2}) = (0.853553) (0.488352) + (0.146447) (- 3.29154) .

⟳Multiply each function value by its weight and add; Gauss–Laguerre is exact for polynomials of degree

\leq 2 n - 1 = 3

= 0.416855 - 0.482052 = - 0.065197

. Step 5. Exact value:

\int_{0}^{\infty} e^{- x} x cos x d x = Re \int_{0}^{\infty} x e^{- (1 - i) x} d x = Re \frac{1}{( 1 - i ) ^{2}} = Re \frac{1}{- 2 i} = 0

. The 2-point estimate

- 0.065

is approximate because

f

is not a polynomial; higher

n

converges to

0

I_{2 -pt} \approx - 0.0652 (exact = 0; dimensionless)

Common Misconceptions

Normalization conventions vary. The convention used here (and by physicists) gives

L_{n} (0) = 1

and the orthonormal form [5]. Some older texts (and Schrödinger's original papers) define Laguerre polynomials differing by a factor of

n!

, and the associated polynomials by repeated differentiation

\frac{d ^{m}}{d x ^{m}} L_{n}

with shifted indices. Always check whether your source defines

L_{n}^{(α)}

via the Rodrigues formula [3] (modern, standard, used by NIST and most quantum mechanics texts) or via

L_{n}^{m} (x) = (- 1)^{m} \frac{d ^{m}}{d x ^{m}} L_{n + m} (x)

(older). They differ by index conventions and constant factors.

Non-integer $n$ does not give a polynomial

One might think equation [4], $x y^{''} + (1 - x) y^{'} + ν y = 0$ , yields a useful function for any $ν$ . But for non-integer $ν$ the power-series solution does not terminate: it is the confluent hypergeometric function $M (- ν, 1, x) =_{1} F_{1} (- ν; 1; x)$ , an infinite series that grows like $e^{x}$ as $x \to \infty$ . This makes $\int_{0}^{\infty} e^{- x} ∣ y ∣^{2} d x$ diverge — the function is not normalizable. Only $ν = n \in Z_{\geq 0}$ truncates the series to a genuine polynomial. This termination requirement is exactly the source of quantum-mechanical energy quantization.

Connections

Laguerre polynomials are one of three classical orthogonal polynomial families, related by limits and the unifying Sturm–Liouville Theory. They connect to Hermite Polynomials (the half-line oscillator versus full-line oscillator) and to Jacobi Polynomials (which limit to Laguerre as one interval endpoint goes to infinity). The associated polynomials are special cases of the Confluent Hypergeometric Function via $L_{n}^{(α)} (x) = (n n + α)_{1} F_{1} (- n; α + 1; x)$ . Their generating function and recurrence link to the broader theory of Generating Functions and Special Functions. In probability, the weight $x^{α} e^{- x}$ is (up to scaling) the Gamma Distribution, so Laguerre polynomials are its orthogonal-polynomial system.

Real-World Applications

Hydrogen atom radial wavefunctions — The radial part of the hydrogen solution is $R_{n ℓ} (r) \propto r^{ℓ} e^{- r / n a_{0}} L_{n - ℓ - 1}^{(2 ℓ + 1)} (2 r / n a_{0})$ ; energy quantization follows from polynomial termination.
Gauss–Laguerre quadrature — Numerical integration of $\int_{0}^{\infty} e^{- x} f (x) d x$ using Laguerre roots as nodes (exact for polynomials up to degree $2 n - 1$ ), used in computational physics and statistics.
Quantum optics / coherent states — Matrix elements of displacement operators and the Wigner functions of Fock states are expressed through Laguerre polynomials.
Laguerre–Gaussian laser modes — The transverse field profiles of cylindrically symmetric laser beams (carrying orbital angular momentum) are $LG_{p ℓ}$ modes built from $L_{p}^{(∣ ℓ ∣)}$ .
Signal processing (Laguerre filters) — Orthonormal Laguerre functions $e^{- x /2} L_{n} (x)$ form bases for system identification and time-domain modeling of linear systems.
Reliability and queueing theory — Expansions of densities over the gamma weight use Laguerre polynomials to model waiting times and failure distributions.
Density functional / electronic structure calculations — Sturmian and Coulomb basis sets exploit Laguerre-type radial functions for atomic computations.

Historical Development

1879 — Edmond Laguerre introduces the polynomials while studying the exponential integral $\int_{x}^{\infty} t^{- 1} e^{- t} d t$ and associated continued fractions.
1880s — Pafnuty Chebyshev and his school develop the general theory of orthogonal polynomials on the half-line, providing the orthogonality framework.
1907 — Erhard Schmidt and contemporaries formalize orthogonal expansions in Hilbert space, giving completeness a rigorous footing.
1926 — Erwin Schrödinger solves the hydrogen atom; the radial equation reveals associated Laguerre polynomials, cementing their physical importance.
1929 — The systematic theory of "classical orthogonal polynomials" (Hermite, Laguerre, Jacobi) is consolidated, notably in work leading to the Bochner classification of which ODEs yield polynomial eigenfunctions.
1934–1939 — Gábor Szegő's monograph Orthogonal Polynomials unifies asymptotics, zeros, and quadrature, becoming the standard reference.
1940s–1950s — Gauss–Laguerre quadrature tables are computed for numerical analysis, accelerated by early electronic computing.
1992 — Laguerre–Gaussian modes are shown by Allen, Beijersbergen, Spreeuw, and Woerdman to carry quantized orbital angular momentum, launching a major field in optics.

Subtleties & Limits

While the polynomials themselves are completely understood, several analytical questions remain delicate. The precise asymptotic distribution of zeros of $L_{n}^{(α)}$ as $n \to \infty$ (and especially the behavior near the "soft edge" governed by Airy functions, or the "hard edge" near $x = 0$ governed by Bessel functions) is an active area connecting to random matrix theory; the largest zero scales like $4 n$ but its fluctuations follow Tracy–Widom statistics. The completeness and convergence of Laguerre series for functions not decaying fast enough is subtle: unlike Fourier series, pointwise and mean convergence require careful weight-dependent conditions ( $L^{p}$ convergence holds only for restricted ranges of $p$ depending on $α$ ). Finally, numerical stability of recurrence [6] for large $n$ and small $x$ remains a practical challenge, motivating ongoing work on stable evaluation algorithms.

Laguerre and Associated Laguerre Polynomials

The Core Insight

Formal Statement

Laguerre Polynomials

Associated (Generalized) Laguerre Polynomials

Laguerre Differential Equation

Orthogonality Relation

Three-Term Recurrence and Generating Function

Why It Holds / Derivation

Orthogonality from Sturm–Liouville structure

Edmond Nicolas Laguerre | 1834–1886

Building Intuition

Worked Examples

Computing $L_3(x)$ and verifying a root

Gauss–Laguerre quadrature of an integral

Common Misconceptions

Non-integer $n$ does not give a polynomial

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads