Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: $$xy'' + (1 - x)y' + ny = 0,\ y = y(x)$$ which is a second-order linear differential equation. This equation has nonsingular solutions only if $n$ is a non-negative integer.

Sometimes the name Laguerre polynomials is used for solutions of $$xy'' + (\alpha + 1 - x)y' + ny = 0~.$$ where $n$ is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).

More generally, a Laguerre function is a solution when $n$ is not necessarily a non-negative integer.

The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form $$\int_0^\infty f(x) e^{-x} \, dx.$$

These polynomials, usually denoted $L_{0}$, $L_{1}$, ..., are a polynomial sequence which may be defined by the Rodrigues formula,

$$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right) =\frac{1}{n!} \left( \frac{d}{dx} -1 \right)^n x^n,$$ reducing to the closed form of a following section.

They are orthogonal polynomials with respect to an inner product $$\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.$$

The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials.

The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator.

Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n ! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)

The first few polynomials
These are the first few Laguerre polynomials:



Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as $$L_0(x) = 1$$ $$L_1(x) = 1 - x$$ and then using the following recurrence relation for any $k ≥ 1$: $$L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}. $$ Furthermore, $$ x L'_n(x) = nL_n (x) - nL_{n-1}(x).$$

In solution of some boundary value problems, the characteristic values can be useful: $$L_{k}(0) = 1, L_{k}'(0) = -k. $$

The closed form is $$L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .$$

The generating function for them likewise follows, $$\sum_{n=0}^\infty t^n L_n(x)= \frac{1}{1-t} e^{-tx/(1-t)}.$$The operator form is $$L_n(x) = \frac{1}{n!}e^x \frac{d^n}{dx^n} (x^n e^{-x}) $$

Polynomials of negative index can be expressed using the ones with positive index: $$L_{-n}(x)=e^xL_{n-1}(-x).$$

Generalized Laguerre polynomials
For arbitrary real α the polynomial solutions of the differential equation $$x\,y'' + \left(\alpha +1 - x\right) y' + n\,y = 0$$ are called generalized Laguerre polynomials, or associated Laguerre polynomials.

One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as $$L^{(\alpha)}_0(x) = 1$$ $$L^{(\alpha)}_1(x) = 1 + \alpha - x$$

and then using the following recurrence relation for any $k ≥ 1$: $$L^{(\alpha)}_{k + 1}(x) = \frac{(2k + 1 + \alpha - x)L^{(\alpha)}_k(x) - (k + \alpha) L^{(\alpha)}_{k - 1}(x)}{k + 1}. $$

The simple Laguerre polynomials are the special case $α = 0$ of the generalized Laguerre polynomials: $$L^{(0)}_n(x) = L_n(x).$$

The Rodrigues formula for them is $$L_n^{(\alpha)}(x) = {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) = \frac{x^{-\alpha}}{n!}\left( \frac{d}{dx}-1\right)^nx^{n+\alpha}.$$

The generating function for them is $$\sum_{n=0}^\infty t^n L^{(\alpha)}_n(x)=  \frac{1}{(1-t)^{\alpha+1}} e^{-tx/(1-t)}.$$

Explicit examples and properties of the generalized Laguerre polynomials

 * Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as $$ L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).$$ where ${n+ \alpha \choose n}$ is a generalized binomial coefficient. When $n$ is an integer the function reduces to a polynomial of degree $n$. It has the alternative expression $$L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x)$$ in terms of Kummer's function of the second kind.
 * The closed form for these generalized Laguerre polynomials of degree $n$ is $$ L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} $$ derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
 * Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let $$D = \frac{d}{dx}$$ and consider the differential operator $$M=xD^2+(\alpha+1)D$$. Then $$\exp(-tM)x^n=(-1)^nt^nn!L^{(\alpha)}_n\left(\frac{x}{t}\right)$$.
 * The first few generalized Laguerre polynomials are:


 * The coefficient of the leading term is $L_{n}^{(k)}(x)$;
 * The constant term, which is the value at 0, is $$L_n^{(\alpha)}(0) = {n+\alpha\choose n} = \frac{\Gamma(n + \alpha + 1)}{n!\, \Gamma(\alpha + 1)};$$

\begin{align} & L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \sin\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha-\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right), \\[6pt] & L_n^{(\alpha)}(-x) = \frac{(n+1)^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-x/2}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} e^{2 \sqrt{x(n+1)}} \cdot\left(1+O\left(\frac{1}{\sqrt{n+1}}\right)\right), \end{align} $$ and summarizing by $$\frac{L_n^{(\alpha)}\left(\frac x n\right)}{n^\alpha}\approx e^{x/ 2n} \cdot \frac{J_\alpha\left(2\sqrt x\right)}{\sqrt x^\alpha},$$ where $$J_\alpha$$ is the Bessel function.
 * If $(−1)^{n}/n !$ is non-negative, then Ln(α) has n real, strictly positive roots (notice that $$\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n$$ is a Sturm chain), which are all in the interval $$\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \, \right].$$
 * The polynomials' asymptotic behaviour for large $n$, but fixed $α$ and $α$, is given by $$

As a contour integral
Given the generating function specified above, the polynomials may be expressed in terms of a contour integral $$L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint_C\frac{e^{-xt/(1-t)}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt,$$ where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1

Recurrence relations
The addition formula for Laguerre polynomials: $$L_n^{(\alpha+\beta+1)}(x+y)= \sum_{i=0}^n L_i^{(\alpha)}(x) L_{n-i}^{(\beta)}(y) .$$

Laguerre's polynomials satisfy the recurrence relations $$L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!},$$ in particular $$L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x)$$ and $$L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x),$$ or $$L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x);$$ moreover $$\begin{align} L_n^{(\alpha)}(x)- \sum_{j=0}^{\Delta-1} {n+\alpha \choose n-j} (-1)^j \frac{x^j}{j!}&= (-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{(n-i){n \choose i}}L_i^{(\alpha+\Delta)}(x)\\[6pt] &=(-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x) \end{align}$$

They can be used to derive the four 3-point-rules $$\begin{align} L_n^{(\alpha)}(x) &= L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j}(-1)^j L_{n-j}^{(\alpha+k)}(x), \\[10pt] n L_n^{(\alpha)}(x) &= (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x), \\[10pt] & \text{or } \\ \frac{x^k}{k!}L_n^{(\alpha)}(x) &= \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x), \\[10pt] n L_n^{(\alpha+1)}(x) &= (n-x) L_{n-1}^{(\alpha+1)}(x) + (n+\alpha)L_{n-1}^{(\alpha)}(x) \\[10pt] x L_n^{(\alpha+1)}(x) &= (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x); \end{align}$$

combined they give this additional, useful recurrence relations$$\begin{align} L_n^{(\alpha)}(x)&= \left(2+\frac{\alpha-1-x}n \right)L_{n-1}^{(\alpha)}(x)- \left(1+\frac{\alpha-1}n \right)L_{n-2}^{(\alpha)}(x)\\[10pt] &= \frac{\alpha+1-x}n L_{n-1}^{(\alpha+1)}(x)- \frac x n L_{n-2}^{(\alpha+2)}(x) \end{align}$$

Since $$L_n^{(\alpha)}(x)$$ is a monic polynomial of degree $$n$$ in $$\alpha$$, there is the partial fraction decomposition $$\begin{align} \frac{n!\,L_n^{(\alpha)}(x)}{(\alpha+1)_n} &= 1- \sum_{j=1}^n (-1)^j \frac{j}{\alpha + j} {n \choose j}L_n^{(-j)}(x) \\ &= 1- \sum_{j=1}^n \frac{x^j}{\alpha + j}\,\,\frac{L_{n-j}^{(j)}(x)}{(j-1)!} \\ &= 1-x \sum_{i=1}^n \frac{L_{n-i}^{(-\alpha)}(x) L_{i-1}^{(\alpha+1)}(-x)}{\alpha +i}. \end{align}$$ The second equality follows by the following identity, valid for integer i and $n$ and immediate from the expression of $$L_n^{(\alpha)}(x)$$ in terms of Charlier polynomials: $$ \frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x).$$ For the third equality apply the fourth and fifth identities of this section.

Derivatives of generalized Laguerre polynomials
Differentiating the power series representation of a generalized Laguerre polynomial $k$ times leads to $$\frac{d^k}{d x^k} L_n^{(\alpha)} (x) = \begin{cases} (-1)^k L_{n-k}^{(\alpha+k)}(x) & \text{if } k\le n, \\ 0 & \text{otherwise.} \end{cases}$$

This points to a special case ($x > 0$) of the formula above: for integer $α = 0$ the generalized polynomial may be written $$L_n^{(k)}(x)=(-1)^k\frac{d^kL_{n+k}(x)}{dx^k},$$ the shift by $k$ sometimes causing confusion with the usual parenthesis notation for a derivative.

Moreover, the following equation holds: $$\frac{1}{k!} \frac{d^k}{d x^k} x^\alpha L_n^{(\alpha)} (x) = {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x),$$ which generalizes with Cauchy's formula to $$L_n^{(\alpha')}(x) = (\alpha'-\alpha) {\alpha'+ n \choose \alpha'-\alpha} \int_0^x \frac{t^\alpha (x-t)^{\alpha'-\alpha-1}}{x^{\alpha'}} L_n^{(\alpha)}(t)\,dt.$$

The derivative with respect to the second variable $α$ has the form, $$\frac{d}{d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}.$$ This is evident from the contour integral representation below.

The generalized Laguerre polynomials obey the differential equation $$x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0,$$ which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

$$x L_n^{[k] \prime\prime}(x) + (k+1-x)L_n^{[k]\prime}(x) + (n-k) L_n^{[k]}(x)=0,$$ where $$L_n^{[k]}(x)\equiv\frac{d^kL_n(x)}{dx^k}$$ for this equation only.

In Sturm–Liouville form the differential equation is

$$-\left(x^{\alpha+1} e^{-x}\cdot L_n^{(\alpha)}(x)^\prime\right)' = n\cdot x^\alpha e^{-x}\cdot L_n^{(\alpha)}(x),$$

which shows that $α = k$ is an eigenvector for the eigenvalue $n$.

Orthogonality
The generalized Laguerre polynomials are orthogonal over $[0, ∞)$ with respect to the measure with weighting function $L(α) n$:

$$\int_0^\infty x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!} \delta_{n,m},$$

which follows from

$$\int_0^\infty x^{\alpha'-1} e^{-x} L_n^{(\alpha)}(x)dx= {\alpha-\alpha'+n \choose n} \Gamma(\alpha').$$

If $$\Gamma(x,\alpha+1,1)$$ denotes the gamma distribution then the orthogonality relation can be written as

$$\int_0^{\infty} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)\Gamma(x,\alpha+1,1) dx={n+ \alpha \choose n}\delta_{n,m},$$

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)

$$\begin{align} K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)} \sum_{i=0}^n \frac{L_i^{(\alpha)}(x) L_i^{(\alpha)}(y)}\\[4pt] & =\frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha)}(x) L_{n+1}^{(\alpha)}(y) - L_{n+1}^{(\alpha)}(x) L_n^{(\alpha)}(y)}{\frac{x-y}{n+1} {n+\alpha \choose n}} \\[4pt] &= \frac{1}{\Gamma(\alpha+1)}\sum_{i=0}^n \frac{x^i}{i!} \frac{L_{n-i}^{(\alpha+i)}(x) L_{n-i}^{(\alpha+i+1)}(y)}; \end{align}$$

recursively

$$K_n^{(\alpha)}(x,y)=\frac{y}{\alpha+1} K_{n-1}^{(\alpha+1)}(x,y)+ \frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha+1)}(x) L_n^{(\alpha)}(y)}.$$

Moreover,

$$y^\alpha e^{-y} K_n^{(\alpha)}(\cdot, y) \to \delta(y- \cdot).$$

Turán's inequalities can be derived here, which is $$L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{n{n\choose k}} L_k^{(\alpha-1)}(x)^2>0.$$

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

$$\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)} (x)\right]^2 dx= \frac{(n+\alpha)!}{n!}(2n+\alpha+1).$$

Series expansions
Let a function have the (formal) series expansion $$f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).$$

Then $$f_i^{(\alpha)}=\int_0^\infty \frac{L_i^{(\alpha)}(x)} \cdot \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \cdot f(x) \,dx .$$

The series converges in the associated Hilbert space $x^{α} e^{−x}$ if and only if

$$\| f \|_{L^2}^2 := \int_0^\infty \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} | f(x)|^2 \, dx = \sum_{i=0}^\infty {i+\alpha \choose i} |f_i^{(\alpha)}|^2 < \infty. $$

Further examples of expansions
Monomials are represented as $$\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x),$$ while binomials have the parametrization $${n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).$$

This leads directly to $$e^{-\gamma x}= \sum_{i=0}^\infty \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x) \qquad \text{convergent iff } \Re(\gamma) > -\tfrac{1}{2}$$ for the exponential function. The incomplete gamma function has the representation $$\Gamma(\alpha,x)=x^\alpha e^{-x} \sum_{i=0}^\infty \frac{L_i^{(\alpha)}(x)}{1+i} \qquad \left(\Re(\alpha)>-1, x > 0\right).$$

In quantum mechanics
In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.

Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.

Multiplication theorems
Erdélyi gives the following two multiplication theorems

$$\begin{align} & t^{n+1+\alpha} e^{(1-t) z} L_n^{(\alpha)}(z t)=\sum_{k=n}^\infty {k \choose n}\left(1-\frac 1 t\right)^{k-n} L_k^{(\alpha)}(z), \\[6pt] & e^{(1-t)z} L_n^{(\alpha)}(z t)=\sum_{k=0}^\infty \frac{(1-t)^k z^k}{k!}L_n^{(\alpha+k)}(z). \end{align}$$

Relation to Hermite polynomials
The generalized Laguerre polynomials are related to the Hermite polynomials: $$\begin{align} H_{2n}(x) &= (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2) \\[4pt] H_{2n+1}(x) &= (-1)^n 2^{2n+1} n! x L_n^{(1/2)} (x^2) \end{align}$$ where the $L^{2}[0, ∞)$ are the Hermite polynomials based on the weighting function $H_{n}(x)$, the so-called "physicist's version."

Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Relation to hypergeometric functions
The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as $$L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)$$ where $$(a)_n$$ is the Pochhammer symbol (which in this case represents the rising factorial).

Hardy–Hille formula
The generalized Laguerre polynomials satisfy the Hardy–Hille formula $$\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(\alpha + 1;\frac{xyt}{(1-t)^2}\right),$$ where the series on the left converges for $$\alpha>-1$$ and $$|t|<1$$. Using the identity $$\,_0F_1(\alpha + 1;z)=\,\Gamma(\alpha + 1) z^{-\alpha/2} I_\alpha\left(2\sqrt{z}\right),$$ (see generalized hypergeometric function), this can also be written as $$\sum_{n=0}^\infty \frac{n!}{\Gamma(1+\alpha+n)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y) t^n = \frac{1}{(xyt)^{\alpha/2}(1-t)}e^{-(x+y)t/(1-t)} I_\alpha \left(\frac{2\sqrt{xyt}}{1-t}\right).$$ This formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.

Physics Convention
The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals. The convention used throughout this article expresses the generalized Laguerre polynomials as

$$L_n^{(\alpha)}(x) = \frac{\Gamma(\alpha + n + 1)}{\Gamma(\alpha + 1) n!} \,_1F_1(-n; \alpha + 1; x),$$

where $$\,_1F_1(a;b;x)$$ is the confluent hypergeometric function. In the physics literature, the generalized Laguerre polynomials are instead defined as

$$\bar{L}_n^{(\alpha)}(x) = \frac{\left[\Gamma(\alpha + n + 1)\right]^2}{\Gamma(\alpha + 1)n!} \,_1F_1(-n; \alpha + 1; x).$$

The physics version is related to the standard version by

$$\bar{L}_n^{(\alpha)}(x) = (n+\alpha)! L_n^{(\alpha)}(x).$$

There is yet another, albeit less frequently used, convention in the physics literature

$$\tilde{L}_n^{(\alpha)}(x) = (-1)^{\alpha}\bar{L}_{n-\alpha}^{(\alpha)}.$$

Umbral Calculus Convention
Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for $$D/(D-I)$$ when multiplied by $$n!$$. In Umbral Calculus convention, the default Laguerre polynomials are defined to be$$\mathcal L_n(x) = n!L_n^{(-1)}(x) = \sum_{k=0}^n L(n,k) (-x)^k$$where $L(n,k) = \binom{n-1}{k-1} \frac{n!}{k!}$ are the signless Lah numbers. $(\mathcal L_n(x))_{n\in\N}$ is a sequence of polynomials of binomial type, ie they satisfy$$\mathcal L_n(x+y) = \sum_{k=0}^n \binom{n}{k} \mathcal L_k(x) \mathcal L_{n-k}(y)$$