Laguerre and Associated Laguerre Polynomials

⚡ Key Takeaways

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 $L_n^{(\alpha)}(x)$ generalize these with weight $x^\alpha e^{-x}$ . 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 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 $x^\alpha e^{-x}$ because they are eigenfunctions of a self-adjoint (Sturm–Liouville) operator. Orthogonality, recurrence, and the Rodrigues formula are all manifestations of this single structure.

DefinitionLaguerre Polynomials

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

1

\quad L_n(x) = \frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}\left(e^{-x} x^{n}\right).

Derivation hint: 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

2

\quad L_n(x) = \sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^{k}}{k!}\, x^{k},

Derivation hint: Expand the Rodrigues formula via Leibniz's rule and use $\frac{d^n}{dx^n}(x^{n+k-...})$ ; collect terms. where $x \in [0,\infty)$ , $n \in \mathbb{Z}_{\ge 0}$ , $\binom{n}{k}$ is the binomial coefficient, and $L_0(x) = 1$ .

DefinitionAssociated (Generalized) Laguerre Polynomials

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

3

\quad L_n^{(\alpha)}(x) = \frac{x^{-\alpha}e^{x}}{n!}\frac{d^{n}}{dx^{n}}\left(e^{-x}x^{n+\alpha}\right) = \sum_{k=0}^{n}(-1)^{k}\binom{n+\alpha}{n-k}\frac{x^{k}}{k!}.

Derivation hint: Generalize the Rodrigues formula by inserting $x^\alpha$ into the weight; expand with Leibniz's rule, identifying $\binom{n+\alpha}{n-k}=\frac{\Gamma(n+\alpha+1)}{\Gamma(k+\alpha+1)(n-k)!}$ . When $\alpha = 0$ these reduce to the ordinary Laguerre polynomials: $L_n^{(0)}(x) = L_n(x)$ . The symbol $\Gamma$ denotes the Gamma function, extending the factorial to non-integer arguments.

TheoremLaguerre Differential Equation

$L_n^{(\alpha)}(x)$ is the polynomial solution of the second-order linear ODE

4

\quad x\,y'' + (\alpha + 1 - x)\,y' + n\,y = 0,

Derivation hint: Substitute the series for $L_n^{(\alpha)}$ and verify term by term, or recognize it as the eigenvalue equation of the Laguerre operator with eigenvalue $n$ . with eigenvalue $n$ . For $\alpha = 0$ 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 $n$ .

TheoremOrthogonality Relation

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

5

\quad \int_{0}^{\infty} x^{\alpha} e^{-x}\, L_m^{(\alpha)}(x)\, L_n^{(\alpha)}(x)\, dx = \frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},

Derivation hint: Cast equation [4] into Sturm–Liouville form $\frac{d}{dx}(x^{\alpha+1}e^{-x}y') = -n\,x^\alpha e^{-x} y$ ; self-adjointness gives $\delta_{mn}$ , and the norm follows from the Rodrigues formula via repeated integration by parts. where $\delta_{mn}$ is the Kronecker delta. For $\alpha = 0$ the norm is exactly $1$ , so the $L_n(x)$ are orthonormal under $e^{-x}$ .

TheoremThree-Term Recurrence and Generating Function

The polynomials satisfy the recurrence

6

\quad (n+1)\,L_{n+1}^{(\alpha)}(x) = (2n+\alpha+1-x)\,L_n^{(\alpha)}(x) - (n+\alpha)\,L_{n-1}^{(\alpha)}(x),

Derivation hint: 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

7

\quad \sum_{n=0}^{\infty} L_n^{(\alpha)}(x)\, t^{n} = \frac{1}{(1-t)^{\alpha+1}}\exp\!\left(\frac{-x\,t}{1-t}\right), \qquad |t|<1.

Derivation hint: Expand the exponential and the binomial $(1-t)^{-(\alpha+1)}$ as power series in $t$ , collect the coefficient of $t^n$ , and match equation [3].

ProofOrthogonality from Sturm–Liouville structure

Step 1. Rewrite Laguerre's equation [4] in self-adjoint (Sturm–Liouville) form. Multiply by the integrating factor $x^{\alpha}e^{-x}$ :

8

\quad \frac{d}{dx}\!\left(x^{\alpha+1}e^{-x}\,y'\right) = -n\, x^{\alpha} e^{-x}\, y.

Derivation hint: Note $\frac{d}{dx}(x^{\alpha+1}e^{-x}) = x^{\alpha}e^{-x}(\alpha+1-x)$ , which reproduces the first-order coefficient in [4].

Step 2. Let $L_m^{(\alpha)}$ , $L_n^{(\alpha)}$ correspond to eigenvalues $m, n$ . Write [8] for each, multiply the $m$ -equation by $L_n^{(\alpha)}$ and the $n$ -equation by $L_m^{(\alpha)}$ , subtract, and integrate over $[0,\infty)$ .

Step 3. The left side becomes a total derivative of $x^{\alpha+1}e^{-x}(L_m' L_n - L_n' L_m)$ , which vanishes at $x=0$ (the factor $x^{\alpha+1}$ kills it for $\alpha>-1$ ) and at $x=\infty$ (the factor $e^{-x}$ dominates). Hence the boundary term is zero.

Step 4. What remains is $(n-m)\int_0^\infty x^\alpha e^{-x} L_m^{(\alpha)} L_n^{(\alpha)} dx = 0$ . For $m \ne n$ 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 $e^{-x}x^{n+\alpha}$ ; boundary terms vanish, and the surviving integral is $\frac{1}{n!}\int_0^\infty x^{n+\alpha}e^{-x}dx \cdot (\text{leading coefficient factor})$ . Evaluating with $\int_0^\infty x^{n+\alpha}e^{-x}dx = \Gamma(n+\alpha+1)$ yields the norm $\Gamma(n+\alpha+1)/n!$ in [5]. $\blacksquare$

∎

EN

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 $\int_x^\infty \frac{e^{-t}}{t}dt$ 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.

🧠 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 $\alpha$ controls how strongly the weight is pushed away from $x=0$ : larger $\alpha$ means the cloud thins out at the origin (the $x^\alpha$ factor suppresses small $x$ ), shifting where the polynomials "live."

ExampleComputing

L_3(x)

and verifying a root

Given: Use the explicit formula [2] for $n=3$ , $\alpha=0$ .

Step 1. Apply $L_n(x) = \sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{k!}x^k$ with $n=3$ :

$k=0$ : $\binom{3}{0}\frac{1}{0!} = 1$
$k=1$ : $\binom{3}{1}\frac{-1}{1!} = -3$
$k=2$ : $\binom{3}{2}\frac{1}{2!} = 3\cdot\tfrac12 = \tfrac32$
$k=3$ : $\binom{3}{3}\frac{-1}{3!} = -\tfrac16$

Step 2. Assemble:

9

\quad L_3(x) = 1 - 3x + \tfrac{3}{2}x^2 - \tfrac{1}{6}x^3.

Derivation hint: 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: $xy'' + (1-x)y' + 3y$ with $y=L_3$ gives $0$ identically (one may verify the cubic coefficient: $y''=3-x$ , $y'=-3+3x-\tfrac12 x^2$ ).

\boxed{L_3(x) = 1 - 3x + \tfrac{3}{2}x^2 - \tfrac{1}{6}x^3 \quad(\text{dimensionless})}

ExampleGauss–Laguerre quadrature of an integral

Given: Approximate $I = \int_0^\infty e^{-x} x\,\cos x\, dx$ using 2-point Gauss–Laguerre quadrature. The nodes are the roots of $L_2(x) = 1 - 2x + \tfrac12 x^2$ , 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 $\tfrac12 x^2 - 2x + 1 = 0 \Rightarrow x^2 - 4x + 2 = 0 \Rightarrow x = 2 \pm \sqrt2$ . 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 $\int_0^\infty e^{-x}f(x)dx \approx \sum_i w_i f(x_i)$ 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:

10

\quad I \approx w_1 f(x_1) + w_2 f(x_2) = (0.853553)(0.488352) + (0.146447)(-3.29154).

Derivation hint: Multiply each function value by its weight and add; Gauss–Laguerre is exact for polynomials of degree $\le 2n-1 = 3$ . $= 0.416855 - 0.482052 = -0.065197$ .

Step 5. Exact value: $\int_0^\infty e^{-x}x\cos x\,dx = \operatorname{Re}\!\int_0^\infty x e^{-(1-i)x}dx = \operatorname{Re}\frac{1}{(1-i)^2} = \operatorname{Re}\frac{1}{-2i} = 0$ . The 2-point estimate $-0.065$ is approximate because $f$ is not a polynomial; higher $n$ converges to $0$ .

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

⚠️ Common mistake

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 $\frac{d^m}{dx^m}L_{n}$ with shifted indices. Always check whether your source defines $L_n^{(\alpha)}$ via the Rodrigues formula [3] (modern, standard, used by NIST and most quantum mechanics texts) or via $L_n^m(x) = (-1)^m \frac{d^m}{dx^m}L_{n+m}(x)$ (older). They differ by index conventions and constant factors.

CounterexampleNon-integer

n

does not give a polynomial

One might think equation [4], $xy'' + (1-x)y' + \nu y = 0$ , yields a useful function for any $\nu$ . But for non-integer $\nu$ the power-series solution does not terminate: it is the confluent hypergeometric function $M(-\nu, 1, x) = {}_1F_1(-\nu;1;x)$ , an infinite series that grows like $e^x$ as $x\to\infty$ . This makes $\int_0^\infty e^{-x}|y|^2 dx$ diverge — the function is not normalizable. Only $\nu = n \in \mathbb{Z}_{\ge0}$ truncates the series to a genuine polynomial. This termination requirement is exactly the source of quantum-mechanical energy quantization.

connection

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 $L_n^{(\alpha)}(x) = \binom{n+\alpha}{n}\,{}_1F_1(-n;\alpha+1;x)$ . Their generating function and recurrence link to the broader theory of Generating Functions and Special Functions. In probability, the weight $x^\alpha e^{-x}$ is (up to scaling) the Gamma Distribution, so Laguerre polynomials are its orthogonal-polynomial system.

🔬 Where this appears

⏳ History

- 1879 — Edmond Laguerre introduces the polynomials while studying the exponential integral

\int_x^\infty t^{-1}e^{-t}dt

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.

open-problem

While the polynomials themselves are completely understood, several analytical questions remain delicate. The precise asymptotic distribution of zeros of $L_n^{(\alpha)}$ 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 $4n$ 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 $\alpha$ ). 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

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also