View source for Laguerre polynomials

{{Short description|Sequence of differential equation solutions}}
[[File:Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i.svg|alt=Complex color plot of {{math|1=''L''<sub>&minus;1/9</sub>(''z''<sup>4</sup>) from &minus;2&minus;2''i'' to 2+2''i''}}|thumb|Complex color plot of {{math|1=''L''<sub>&minus;1/9</sub>(''z''<sup>4</sup>) from &minus;2&minus;2''i'' to 2+2''i''}}]]
In [[mathematics]], the '''Laguerre polynomials''', named after [[Edmond Laguerre]] (1834–1886), are nontrivial solutions of '''Laguerre's differential equation:'''
<math display="block">xy'' + (1 - x)y' + ny = 0,\     
y = y(x)</math>
which is a second-order [[linear differential equation]]. This equation has [[nonsingular solution]]s only if {{mvar|n}} is a non-negative integer.

Sometimes the name '''Laguerre polynomials''' is used for solutions of
<math display="block">xy'' + (\alpha + 1 - x)y' + ny = 0~.</math>
where {{mvar|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<ref>{{cite journal|title=Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries|author=N. Sonine|journal=[[Math. Ann.]]|date=1880|volume=16| issue=1|pages=1–80|doi=10.1007/BF01459227|s2cid=121602983|url=http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0016&DMDID=dmdlog8}}</ref> [[Nikolay Yakovlevich Sonin]]).

More generally, a '''Laguerre function''' is a solution when {{mvar|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
<math display="block">\int_0^\infty f(x) e^{-x} \, dx.</math>

These polynomials, usually denoted {{math|''L''<sub>0</sub>}},&nbsp;{{math|''L''<sub>1</sub>}},&nbsp;..., are a [[polynomial sequence]] which may be defined by the [[Rodrigues formula#Rodrigues formula|Rodrigues formula]],

<math display="block">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,</math>
reducing to the closed form of a following section.

They are [[orthogonal polynomials]] with respect to an [[inner product]]
<math display="block">\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.</math>

The [[rook polynomial]]s 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 [[Phase space formulation#Simple harmonic oscillator|quantum mechanics in phase space]]. They further enter in the quantum mechanics of the [[Morse potential]] and of the [[Quantum harmonic oscillator#Example: 3D isotropic harmonic oscillator|3D isotropic harmonic oscillator]].

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

== Recursive definition, closed form, and generating function ==
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
<math display="block">L_0(x) = 1</math>
<math display="block">L_1(x) = 1 - x</math>
and then using the following [[Orthogonal polynomials#Recurrence relations|recurrence relation]] for any {{math|''k'' ≥ 1}}:
<math display="block">L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}. </math>
Furthermore,
<math display="block">  x L'_n(x) = nL_n (x) - nL_{n-1}(x).</math>

In solution of some boundary value problems, the characteristic values can be useful:
<math display="block">L_{k}(0) = 1, L_{k}'(0) = -k.  </math>

The '''[[Closed-form expression|closed form]]''' is
<math display="block">L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .</math>

The [[generating function]] for them likewise follows, 
<math display="block">\sum_{n=0}^\infty t^n L_n(x)=  \frac{1}{1-t} e^{-tx/(1-t)}.</math>The operator form is
<math display="block">L_n(x) = \frac{1}{n!}e^x \frac{d^n}{dx^n} (x^n e^{-x}) </math>

Polynomials of negative index can be expressed using the ones with positive index:
<math display="block">L_{-n}(x)=e^xL_{n-1}(-x).</math>

{| class="wikitable" style="margin:0.5em auto"
|+A table of the Laguerre polynomials
|-
! width="20%" | ''n''
! <math>L_n(x)\,</math>
|-
| align="center" | 0 || <math>1\,</math>
|-
| align="center" | 1 || <math>-x+1\,</math>
|-
| align="center" | 2
| <math> \tfrac{1}{2} (x^2-4x+2) \,</math>
|-
| align="center" | 3
| <math>\tfrac{1}{6} (-x^3+9x^2-18x+6) \,</math>
|-
| align="center" | 4
| <math>\tfrac{1}{24} (x^4-16x^3+72x^2-96x+24) \,</math>
|-
| align="center" | 5
| <math>\tfrac{1}{120} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math>
|-
| align="center" | 6
| <math>\tfrac{1}{720} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math>
|-
| align="center" | 7
| <math>\tfrac{1}{5040} (-x^7+49x^6-882x^5+7350x^4-29400x^3+52920x^2-35280x+5040) \,</math>
|-
| align="center" | 8
| <math>\tfrac{1}{40320} (x^8-64x^7+1568x^6-18816x^5+117600x^4-376320x^3+564480x^2-322560x+40320) \,</math>
|-
| align="center" | 9
| <math>\tfrac{1}{362880} (-x^9+81x^8-2592x^7+42336x^6-381024x^5+1905120x^4-5080320x^3+6531840x^2-3265920x+362880) \,</math>
|-
| align="center" | 10
| <math>\tfrac{1}{3628800} (x^{10}-100x^9+4050x^8-86400x^7+1058400x^6-7620480x^5+31752000x^4-72576000x^3+81648000x^2-36288000x+3628800) \,</math>
|-
| align="center" | ''n''
| <math>\tfrac{1}{n!} ((-x)^n + n^2(-x)^{n-1} + \dots + \binom{n}{k}^2 {k!} (-x)^{n -k}+ \dots + n({n!})(-x) + n!) \,</math>
|}

[[Image:Laguerre poly.svg|thumb|center|600px|The first six Laguerre polynomials.]]

== Generalized Laguerre polynomials ==
For arbitrary real α the polynomial solutions of the differential equation<ref>A&S p. 781</ref>
<math display="block">x\,y'' + \left(\alpha +1 - x\right) y' + n\,y = 0</math>
are called '''generalized Laguerre polynomials''', or '''associated Laguerre polynomials'''.

One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as
<math display="block">L^{(\alpha)}_0(x) = 1</math>
<math display="block">L^{(\alpha)}_1(x) = 1 + \alpha - x</math>

and then using the following [[Orthogonal polynomials#Recurrence relations|recurrence relation]] for any {{math|''k'' ≥ 1}}:
<math display="block">L^{(\alpha)}_{k + 1}(x) = \frac{(2k + 1 + \alpha - x)L^{(\alpha)}_k(x) - (k + \alpha) L^{(\alpha)}_{k - 1}(x)}{k + 1}. </math>

The simple Laguerre polynomials are the special case {{math|1=''α'' = 0}} of the generalized Laguerre polynomials:
<math display="block">L^{(0)}_n(x) = L_n(x).</math>

The [[Rodrigues formula]] for them is
<math display="block">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}.</math>

The [[generating function]] for them is
<math display="block">\sum_{n=0}^\infty  t^n L^{(\alpha)}_n(x)=  \frac{1}{(1-t)^{\alpha+1}} e^{-tx/(1-t)}.</math>
[[File:Zugeordnete Laguerre-Polynome.svg|thumb|center|600px|The first few generalized Laguerre polynomials, {{math|''L<sub>n</sub>''<sup>(''k'')</sup>(''x'')}}]]

=== Properties ===
* Laguerre functions are defined by [[confluent hypergeometric function]]s and Kummer's transformation as<ref>A&S p. 509</ref> <math display="block"> L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).</math> where <math display="inline">{n+ \alpha \choose n}</math> is a generalized [[binomial coefficient]]. When {{mvar|n}} is an integer the function reduces to a polynomial of degree {{mvar|n}}. It has the alternative expression<ref>A&S p. 510</ref> <math display="block">L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x)</math> in terms of [[confluent hypergeometric function|Kummer's function of the second kind]].
* The closed form for these generalized Laguerre polynomials of degree {{mvar|n}} is<ref>A&S p. 775</ref> <math display="block">
 L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} 
</math> derived by applying [[Leibniz rule (generalized product rule)|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  <math>D = \frac{d}{dx}</math> and consider the differential operator <math>M=xD^2+(\alpha+1)D</math>. Then <math>\exp(-tM)x^n=(-1)^nt^nn!L^{(\alpha)}_n\left(\frac{x}{t}\right)</math>.{{citation needed|date=April 2023}}
* The first few generalized Laguerre polynomials are:

{| class="wikitable" style="margin:0.5em auto"
|-
! width="20%"| ''n''
! <math>L_n^{(\alpha)}(x)\,</math>
|-
| align="center" | 0
| <math>1\,</math>
|-
| align="center" | 1
| <math>-x+\alpha +1\,</math>
|-
| align="center" | 2
| <math> \tfrac{1}{2} (x^2-2\left( \alpha +2 \right) x+\left( \alpha +1 \right) \left( \alpha +2 \right)) \,</math>
|-
| align="center" | 3
| <math>\tfrac{1}{6} (-x^3+3\left( \alpha +3 \right) x^2-3\left( \alpha +2 \right) \left( \alpha +3 \right) x+\left( \alpha +1 \right) \left( \alpha +2 \right) \left( \alpha +3 \right)) \,</math>
|-
| align="center" | 4
| <math>\tfrac{1}{24} (x^4-4\left( \alpha +4 \right) x^3+6\left( \alpha +3 \right) \left( \alpha +4 \right) x^2-4\left( \alpha +2 \right) \cdots \left( \alpha +4 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +4 \right)) \,</math>
|-
| align="center" | 5
| <math>\tfrac{1}{120} (-x^5+5\left( \alpha +5 \right) x^4-10\left( \alpha +4 \right) \left( \alpha +5 \right) x^3+10\left( \alpha +3 \right) \cdots \left( \alpha +5 \right) x^2-5\left( \alpha +2 \right) \cdots \left( \alpha +5 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +5 \right)) \,</math>
|-
| align="center" | 6
| <math>\tfrac{1}{720} (x^6-6\left( \alpha +6 \right) x^5+15\left( \alpha +5 \right) \left( \alpha +6 \right) x^4-20\left( \alpha +4 \right) \cdots \left( \alpha +6 \right) x^3+15\left( \alpha +3 \right) \cdots \left( \alpha +6 \right) x^2-6\left( \alpha +2 \right) \cdots \left( \alpha +6 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +6 \right)) \,</math>
|-
| align="center" | 7
| <math>\tfrac{1}{5040} (-x^7+7\left( \alpha +7 \right) x^6-21\left( \alpha +6 \right) \left( \alpha +7 \right) x^5+35\left( \alpha +5 \right) \cdots \left( \alpha +7 \right) x^4-35\left( \alpha +4 \right) \cdots \left( \alpha +7 \right) x^3+21\left( \alpha +3 \right) \cdots \left( \alpha +7 \right) x^2-7\left( \alpha +2 \right) \cdots \left( \alpha +7 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +7 \right)) \,</math>
|-
| align="center" | 8
| <math>\tfrac{1}{40320} (x^8-8\left( \alpha +8 \right) x^7+28\left( \alpha +7 \right) \left( \alpha +8 \right) x^6-56\left( \alpha +6 \right) \cdots \left( \alpha +8 \right) x^5+70\left( \alpha +5 \right) \cdots \left( \alpha +8 \right) x^4-56\left( \alpha +4 \right) \cdots \left( \alpha +8 \right) x^3+28\left( \alpha +3 \right) \cdots \left( \alpha +8 \right) x^2-8\left( \alpha +2 \right) \cdots \left( \alpha +8 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +8 \right)) \,</math>
|-
| align="center" | 9
| <math>\tfrac{1}{362880} (-x^9+9\left( \alpha +9 \right) x^8-36\left( \alpha +8 \right) \left( \alpha +9 \right) x^7+84\left( \alpha +7 \right) \cdots \left( \alpha +9 \right) x^6-126\left( \alpha +6 \right) \cdots \left( \alpha +9 \right) x^5+126\left( \alpha +5 \right) \cdots \left( \alpha +9 \right) x^4-84\left( \alpha +4 \right) \cdots \left( \alpha +9 \right) x^3+36\left( \alpha +3 \right) \cdots \left( \alpha +9 \right) x^2-9\left( \alpha +2 \right) \cdots \left( \alpha +9 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +9 \right)) \,</math>
|-
| align="center" | 10
| <math>\tfrac{1}{3628800} (x^{10}-10\left( \alpha +10 \right) x^9+45\left( \alpha +9 \right) \left( \alpha +10 \right) x^8-120\left( \alpha +8 \right) \cdots \left( \alpha +10 \right) x^7+210\left( \alpha +7 \right) \cdots \left( \alpha +10 \right) x^6-252\left( \alpha +6 \right) \cdots \left( \alpha +10 \right) x^5+210\left( \alpha +5 \right) \cdots \left( \alpha +10 \right) x^4-120\left( \alpha +4 \right) \cdots \left( \alpha +10 \right) x^3+45\left( \alpha +3 \right) \cdots \left( \alpha +10 \right) x^2-10\left( \alpha +2 \right) \cdots \left( \alpha +10 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +10 \right)) \,</math>
|}

* The [[coefficient]] of the leading term is {{math|(−1)<sup>''n''</sup>/''n''<nowiki>!</nowiki>}};
* <!-- \frac{n^\alpha}{\Gamma(\alpha+1)} + O\left(n^{\alpha-1}\right);</math> -->The [[constant term]], which is the value at&nbsp;0, is <math display="block">L_n^{(\alpha)}(0) = {n+\alpha\choose n} = \frac{\Gamma(n + \alpha + 1)}{n!\, \Gamma(\alpha + 1)};</math>
* The [[discriminant]] is<ref name=":2">{{Cite web |title=DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.16 |website=dlmf.nist.gov}}</ref><math display="block">\operatorname{Disc}\left(L_n^{(\alpha)}\right)=\prod_{j=1}^n j^{j-2 n+2}(j+\alpha)^{j-1}</math>

=== As a contour integral ===
Given the generating function specified above, the polynomials may be expressed in terms of a [[contour integral]]
<math display="block">L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint_C\frac{e^{-xt/(1-t)}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt,</math>
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:<ref>{{Cite web |title=DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.18 |access-date=2025-03-18 |website=dlmf.nist.gov}}</ref>
<math display="block">L^{(\alpha_{1}+\dots+\alpha_{r}+r-1)}_{n}\left(x_{1}+\dots+x_{r}\right)=\sum_{m_{1}+\dots+m_{r}=n}L^{(\alpha_{1})}_{m_{1}}\left(x_{1}\right)\cdots L^{(\alpha_{r})}_{m_{r}}\left(x_{r}\right).</math>Laguerre's polynomials satisfy the recurrence relations
<math display="block">L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!},</math>
in particular
<math display="block">L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x)</math>
and
<math display="block">L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x),</math>
or
<math display="block">L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x);</math>
moreover
<math display="block">\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+\alpha \choose n-\Delta-i}}{(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+\alpha-i-1 \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x)
\end{align}</math>

They can be used to derive the four 3-point-rules
<math display="block">\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}</math>

combined they give this additional, useful recurrence relations<math display="block">\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}</math>

Since <math>L_n^{(\alpha)}(x)</math> is a [[monic polynomial]] of degree <math>n</math> in <math>\alpha</math>,
there is the [[partial fraction decomposition]]
<math display="block">\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}</math>
The second equality follows by the following identity, valid for integer ''i'' and {{mvar|n}} and immediate from the expression of <math>L_n^{(\alpha)}(x)</math> in terms of [[Charlier polynomials]]:
<math display="block"> \frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x).</math>
For the third equality apply the fourth and fifth identities of this section.

=== Derivatives ===
Differentiating the [[power series]] representation of a generalized Laguerre polynomial {{mvar|k}} times leads to
<math display="block">\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}</math>

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

Moreover, the following equation holds:
<math display="block">\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),</math>
which generalizes with [[Antiderivative#Techniques of integration|Cauchy's formula]] to
<math display="block">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.</math>

The derivative with respect to the second variable {{mvar|α}} has the  form,<ref>{{Cite journal | doi=10.1080/10652469708819127 | title = Identities for families of orthogonal polynomials and special functions| journal=Integral Transforms and Special Functions | volume=5| issue=1–2| pages=69–102|year = 1997|last1 = Koepf|first1 = Wolfram| citeseerx=10.1.1.298.7657}}</ref> 
<math display="block">\frac{d}{d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}.</math>
The generalized Laguerre polynomials obey the differential equation
<math display="block">x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0,</math>
which may be compared with the equation obeyed by the ''k''th derivative of the ordinary Laguerre polynomial,

<math display="block">x L_n^{[k] \prime\prime}(x) + (k+1-x)L_n^{[k]\prime}(x) + (n-k) L_n^{[k]}(x)=0,</math>
where <math>L_n^{[k]}(x)\equiv\frac{d^kL_n(x)}{dx^k}</math> for this equation only.

In [[Sturm–Liouville theory|Sturm–Liouville form]] the differential equation is

<math display="block">-\left(x^{\alpha+1} e^{-x}\cdot L_n^{(\alpha)}(x)^\prime\right)' = n\cdot x^\alpha e^{-x}\cdot L_n^{(\alpha)}(x),</math>

which shows that {{math|''L''{{su|b=''n''|p=''(α)''}}}} is an eigenvector for the eigenvalue {{mvar|n}}.

=== Orthogonality ===
The generalized Laguerre polynomials are [[orthogonal]] over {{closed-open|0, ∞}} with respect to the measure with [[weighting function]] {{math|''x<sup>α</sup>'' ''e''<sup>−''x''</sup>}}:<ref>{{Cite web | url=http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html | title=Associated Laguerre Polynomial}}</ref>

<math display="block">\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},</math>

which follows from

<math display="block">\int_0^\infty x^{\alpha'-1} e^{-x} L_n^{(\alpha)}(x)dx= {\alpha-\alpha'+n \choose n} \Gamma(\alpha').</math>

If <math>\Gamma(x,\alpha+1,1)</math> denotes the [[gamma distribution]] then the orthogonality relation can be written as

<math display="block">\int_0^{\infty} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)\Gamma(x,\alpha+1,1) dx={n+ \alpha \choose n}\delta_{n,m}.</math>

The associated, symmetric kernel polynomial has the representations ([[Christoffel–Darboux formula]]){{citation needed|date=October 2011}}<!--All of these formulas require citations.-->

<math display="block">\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)}{{\alpha+i \choose i}}\\[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)}{{\alpha+n \choose n}{n \choose i}};
\end{align}</math>

recursively

<math display="block">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)}{{\alpha+n \choose n}}.</math>

Moreover,{{clarify|post-text=Limit as n goes to infinity?|date=January 2016}}

<math display="block">y^\alpha e^{-y} K_n^{(\alpha)}(\cdot, y) \to \delta(y- \cdot).</math>

[[Turán's inequalities]] can be derived here, which is
<math display="block">L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{{\alpha+n-1\choose n-k}}{n{n\choose k}} L_k^{(\alpha-1)}(x)^2>0.</math>

The following [[integral]] is needed in the [[quantum mechanics|quantum mechanical]] treatment of the [[Hydrogen atom#Wavefunction|hydrogen atom]],

<math display="block">\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)} (x)\right]^2 dx= \frac{(n+\alpha)!}{n!}(2n+\alpha+1).</math>

=== Series expansions ===
Let a function have the (formal) series expansion
<math display="block">f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).</math>


Then
<math display="block">f_i^{(\alpha)}=\int_0^\infty \frac{L_i^{(\alpha)}(x)}{{i+ \alpha \choose i}} \cdot \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \cdot f(x) \,dx .</math>

The series converges in the associated [[Hilbert space]] {{math|[[Lp space|''L''<sup>2</sup>[0, ∞)]]}} [[if and only if]]

<math display="block">\| 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. </math>

==== Further examples of expansions====
[[Monomial]]s are represented as
<math display="block">\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x),</math>
while [[binomial coefficient|binomials]] have the parametrization
<math display="block">{n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).</math>

This leads directly to
<math display="block">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}</math>
for the exponential function. The [[incomplete gamma function]] has the representation
<math display="block">\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).</math>

=== Asymptotics ===

==== In terms of elementary functions ====
{{Anchor|Perron's formula}}For any fixed positive integer <math>M</math>, fixed real number <math>\alpha</math>, fixed and bounded interval <math>[c, d] \subset (0, + \infty)
</math>, uniformly for <math>x \in [c, d]
</math>, at <math>n \to \infty
</math>:<math display="block">L^{(\alpha)}_{n}\left(x\right)=\frac{n^{\frac{1}{2}\alpha-\frac{1}{4}}{\mathrm{e}}^{\frac{1}{2}x}}{{\pi}^{\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\left(\cos\theta_{n}^{(\alpha)}(x)\left(\sum_{m=0}^{M-1}\frac{a_{m}(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)+\sin\theta_{n}^{(\alpha)}(x)\left(\sum_{m=1}^{M-1}\frac{b_{m}(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)\right)
</math>where <math display="block">
\theta_{n}^{(\alpha)}(x) :=2(nx)^{\frac{1}{2}}-\left(\tfrac{1}{2}\alpha+\tfrac{1}{4}\right)\pi.
</math>and <math>a_0, b_1, a_1, b_2, \dots 
</math> are functions depending on <math>\alpha, x 
</math> but not <math>n 
</math>, and regular for <math>x > 0 
</math>. The first few ones are:<math display="block">\begin{aligned}
& a_0(x)=1 \\
& a_1(x)=0 \\
& b_1(x)=\frac{1}{48 x^{\frac{1}{2}}}\left(4 x^2-24 (\alpha + 1) x+3-12 \alpha^2\right)
\end{aligned}
</math>This is [[Oskar Perron|Perron]]'s formula.<ref name=":1">{{Cite web |title=DLMF: §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.15 |access-date=2025-07-07 |website=dlmf.nist.gov}}</ref><ref>{{Cite journal |last=Perron |first=Oskar |date=1921-01-01 |title=Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters. |url=https://www.degruyterbrill.com/document/doi/10.1515/crll.1921.151.63/html |language=de |volume=1921 |issue=151 |pages=63–78 |doi=10.1515/crll.1921.151.63 |issn=1435-5345}}</ref>{{Reference page|page=78}} There is also a generalization for <math>x \in \mathbb C \setminus [0, \infty) 
</math>.<ref name=":0">Szegő, p. 198.</ref> [[Lipót Fejér|Fejér]]'s formula is a special case of Perron's formula with <math>M = 1</math>.<ref>{{Citation |last=Turán |first=Pál |title=Asymptotikus Értékek Meghatározásáról |date=1970 |work=Leopold Fejér Gesammelte Arbeiten I |pages=445–503 |url=http://link.springer.com/10.1007/978-3-0348-5902-8_31 |access-date=2025-07-07 |place=Basel |publisher=Birkhäuser Basel |language=de |doi=10.1007/978-3-0348-5902-8_31 |isbn=978-3-0348-5903-5|url-access=subscription }}</ref><ref name=":0" /><ref>D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", ''SIAM J. Numer. Anal.'', vol. 46 (2008), no. 6, pp. 3285–3312 {{doi|10.1137/07068031X}}</ref>

==== In terms of Bessel functions ====
The [[Mehler–Heine formula]] states:

: <math>\lim_{n \to \infty} n^{-\alpha}L_n^{(\alpha)}\left(\frac{z^2}{4n}\right)
= \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z),</math>

where <math>J_\alpha</math> is a [[Bessel function#Bessel functions of the first kind: Jα|Bessel function of the first kind]].

See also:.<ref name=":1" />

==== In terms of Airy functions ====
Let <math>\nu  =4 n+2 \alpha+2</math>. Let <math>\operatorname{Ai}</math> be the [[Airy function]]. Let <math>\alpha</math> be arbitrary and real, <math>\epsilon</math> and <math>\omega</math> be positive and fixed.

The [[Plancherel–Rotach asymptotics]] formulas:<ref>Szegő, pp. 200–201</ref><ref name=":1" />

* for <math>x =\nu\cos^2\varphi</math> and <math>\epsilon\leq \varphi \leq \tfrac{\pi}{2} -\epsilon n^{-1/2}</math>, uniformly at <math>n \to \infty
</math>:

:: <math>
e^{-x/2}L^{(\alpha )}_n(x)
=(-1)^{n}(\pi \sin \varphi)^{-1/2}x^{-\alpha/2-1/4}n^{\alpha/2-1/4}
\big\{\sin\left[\left(n+\tfrac{\alpha+1}{2}\right)(\sin 2\varphi-2\varphi)+3\pi/4\right] +(nx)^{-1/2}\mathcal{O}(1)\big\}
</math>

* for <math>x = \nu\cosh^2\varphi</math> and <math>\epsilon\leq \varphi \leq \omega</math>, uniformly at <math>n \to \infty
</math>:

:: <math>
e^{-x/2}L^{(\alpha )}_n(x)
=\tfrac{1}{2}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha/2-1/4}n^{\alpha /2-1/4}
\exp\left[\left(n+\tfrac{\alpha+1}{2}\right)(2\varphi-\sinh 2\varphi)\right]
\{1+\mathcal{O}\left(n^{-1}\right)\}
</math>

* for <math>x =\nu -2(2n/3)^{1/3}t</math> and <math>t</math> complex and bounded, uniformly at <math>n \to \infty
</math>:

:: <math>e^{-x/2}L^{(\alpha)}_n(x)
=(-1)^n\pi^{-1}2^{-\alpha-1/3}3^{1/3}n^{-1/3}
\bigg\{\pi \operatorname{Ai}(-3^{-1/3}t)+\mathcal{O}\left(n^{-2/3}\right)\bigg\}</math>

See DLMF for higher-order terms.<ref name=":1" />

== Zeroes ==

=== Notation ===
<math>j_{\alpha, m}</math> is the <math>m</math>-th positive zero of the [[Bessel function]] <math>J_\alpha(x)</math>.

<math>a_m</math> is the <math>m</math>-th zero of the Airy function <math>\operatorname{Ai}(x)</math>, in descending order: <math>0 > a_1 > a_2 > \cdots</math>.

<math>\nu  =4 n+2 \alpha+2</math>.

If <math>\alpha > -1</math>, then <math>L_n^{(\alpha)}</math> has <math>n</math> real roots. Thus in this section we assume <math>\alpha > -1</math> by default.

<math>x_1 < \dots < x_n</math> are the real roots of <math>L_n^{(\alpha)}</math>.

Note that <math>\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n</math> is a [[Sturm chain]].

=== Inequalities ===
For <math>\alpha > -1</math>, we have these bounds:<ref name=":4">{{Cite journal |last=Driver |first=K. |last2=Jordaan |first2=K. |date=January 2013 |title=Inequalities for Extreme Zeros of Some Classical Orthogonal andq-orthogonal Polynomials |url=https://www.cambridge.org/core/journals/mathematical-modelling-of-natural-phenomena/article/abs/inequalities-for-extreme-zeros-of-some-classical-orthogonal-andqorthogonal-polynomials/23A22C923D65F1DAAE2DF1D2D7EBE8B9 |journal=Mathematical Modelling of Natural Phenomena |language=en |volume=8 |issue=1 |pages=48–59 |doi=10.1051/mmnp/20138103 |issn=0973-5348}}</ref><ref name=":5">{{Cite journal |last=Gatteschi |first=Luigi |date=2002-07-01 |title=Asymptotics and bounds for the zeros of Laguerre polynomials: a survey |url=https://www.sciencedirect.com/science/article/pii/S0377042701005490 |journal=Journal of Computational and Applied Mathematics |series=Selected papers of the Int. Symp. on Applied Mathematics, August 2000, Dalian, China |volume=144 |issue=1 |pages=7–27 |doi=10.1016/S0377-0427(01)00549-0 |issn=0377-0427|url-access=subscription }}</ref><ref name=":2" /><ref>{{Cite journal |last=Dimitrov |first=Dimitar K. |last2=Rafaeli |first2=Fernando R. |date=2009-12-01 |title=Monotonicity of zeros of Laguerre polynomials |url=https://www.sciencedirect.com/science/article/pii/S0377042709001149 |journal=Journal of Computational and Applied Mathematics |series=9th OPSFA Conference |volume=233 |issue=3 |pages=699–702 |doi=10.1016/j.cam.2009.02.038 |issn=0377-0427|url-access=subscription }}</ref>

* <math>x_1 < \frac{(\alpha+1)(\alpha+2)}{n+\alpha+1}</math>
* <math>x_1 < \frac{(\alpha+1)(\alpha+3)}{2 n+\alpha+1}</math>
* <math>x_1 < \frac{(\alpha+1)(\alpha+2)(\alpha+4)(2 n+\alpha+1)}{(\alpha+1)^2(\alpha+2)+n(5 \alpha+11)(n+\alpha+1)}</math>
* <math>x_n \leq 2 n+\alpha-1+2 \sqrt{(n-2)(n+\alpha-1)}</math> when <math>n \geq 2</math>
* <math>x_n > 4n + \alpha - 16 \sqrt{2n}</math>
* <math>x_n > 3n-4</math>
* <math>x_n > 2n + \alpha - 1</math>
* <math>x_n > 2 n+\alpha-2+\sqrt{n^2-2 n+\alpha n+2}</math>
* <math display="block">\begin{aligned}
& (n+2) x_1 &\geq\left(n-1-\sqrt{n^2+(n+2)(\alpha+1)}\right)^2-1 \\
& (n+2) x_n &\leq\left(n-1+\sqrt{n^2+(n+2)(\alpha+1)}\right)^2-1 
\end{aligned}</math>
* <math display="block">\begin{aligned}
x_1 &> \frac 12 \nu - 3 -\sqrt{1+4(n-1)(n+\alpha-1)} \\
x_n &< \frac 12 \nu - 3 +\sqrt{1+4(n-1)(n+\alpha-1)}
\end{aligned}</math>

For fixed <math>k = 1, \dots, n</math>,<ref name=":4" /><ref name=":2" /><ref name=":5" /><math display="block">\begin{aligned}
\nu x_k&>j_{\alpha, k}^2 \\
x_k&< \frac{j_{\alpha, k}^2}{\nu / 2+\sqrt{(\nu / 2)^2-j_{\alpha, k}^2}} \quad \text{ if } \nu / 2 > j_{\alpha, k}\\
x_k &< \left[\nu^{1 / 2}+2^{-1 / 3}\nu^{-1 / 6} a_{n-k+1}\right]^2 \quad \text{ if }|\alpha| \geqslant 1 / 4\\
x_k &< \nu+2^{\frac{2}{3}} a_k \nu^{\frac{1}{3}}+2^{-\frac{2}{3}} a_k^2 \nu^{-\frac{1}{3}}
\end{aligned}</math>For fixed <math>k</math>, we have <math>\lim_{n \to \infty} \nu x_k = j_{\alpha, k}^2</math>, so the first inequality is sharp.

See also.<ref>{{Harvard citation|Szegő|1975|loc=Section 6.21. Inequalities for the zeros of the classical polynomials}}</ref>

=== Electrostatics ===
The zeroes satisfy the '''[[Thomas Joannes Stieltjes|Stieltjes]] relations''':<ref>{{Cite journal |last=Marcellán |first=F. |last2=Martínez-Finkelshtein |first2=A. |last3=Martínez-González |first3=P. |date=2007-10-15 |title=Electrostatic models for zeros of polynomials: Old, new, and some open problems |url=https://www.sciencedirect.com/science/article/pii/S037704270600611X |journal=Journal of Computational and Applied Mathematics |series=Proceedings of The Conference in Honour of Dr. Nico Temme on the Occasion of his 65th birthday |volume=207 |issue=2 |pages=258–272 |doi=10.1016/j.cam.2006.10.020 |issn=0377-0427|arxiv=math/0512293 }}</ref><ref>{{Harvard citation|Szegő|1975|p=|loc=Section 6.7. Electrostatic interpretation of the zeros of the classical polynomials}}</ref><math display="block">\begin{aligned}
\sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{i} - x_{j}} &= \frac 12 \left(1 - \frac{\alpha + 1}{x_i}\right)\\
\sum_{1 \leq j \leq n} \frac{1}{x_{j}} &= \frac{n}{\alpha + 1}\\
\sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^2} &= -\frac{(\alpha + 1)(\alpha + 5)}{12 x_i^2}  + \frac{2n + \alpha + 1}{6x_i}- \frac{1}{12}\\
\sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^3} &= -\frac{(\alpha + 1)(\alpha + 3)}{8 x_i^3} + \frac{2n + \alpha + 1}{8x_i^2}\\
\end{aligned}</math>The first relation can be interpreted physically. Fix an electric particle at origin with charge <math>+\frac{\alpha + 1}{2}</math>, and produce a constant [[electric field]] of strength <math>-\frac 12 </math>. Then, place <math>n </math> electric particles with charge <math>+1 </math>. The first relation states that the zeroes of <math>L_n^{(\alpha)}</math> are the equilibrium positions of the particles.

As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Laguerre polynomials.

The zeroes also satisfy<ref name=":3">{{Cite journal |last=Gupta |first=Dharma P. |last2=Muldoon |first2=Martin E. |date=2007 |title=Inequalities for the smallest zeros of Laguerre polynomials and their -analogues. |url=https://eudml.org/doc/128801 |journal=JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] |volume=8 |issue=1 |pages=Paper No. 24, 7 p., electronic only–Paper No. 24, 7 p., electronic only |issn=1443-5756}}</ref><math display="block">\sum_{i=1}^n \frac{1}{x-x_i }=-\sum_{k=0}^{\infty} S_{k+1} x^k, \quad S_k := \sum_{i = 1}^n x_i^{-k}</math>which allows the following bound<math display="block">S_m^{-1 / m}<x_1<S_m / S_{m+1}, \quad m=1,2, \ldots</math>

=== Limit distribution ===
Let <math>F_n(t) := \frac 1n \#\{i : x_{i} \leq t\}</math> be the [[cumulative distribution function]] for the roots, then we have the limit law<ref>{{Cite journal |last=Gawronski |first=Wolfgang |date=1987-07-01 |title=On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials |url=https://www.sciencedirect.com/science/article/pii/0021904587900207 |journal=Journal of Approximation Theory |volume=50 |issue=3 |pages=214–231 |doi=10.1016/0021-9045(87)90020-7 |issn=0021-9045|url-access=subscription }}</ref><math display="block">\lim_{n \to \infty} F_n(4n t) = \frac 2\pi \int_{0}^t \sqrt{\frac{1-s}{s}} ds \quad \forall t \in (0, 1] </math>which can be interpreted as the limit distribution of the [[Wishart ensemble]] spectrum.

For fixed <math>\alpha > -1</math> and fixed <math>k</math>, as <math>n \to \infty</math>,<ref name=":5" /><math display="block">\begin{aligned}
x_{n+1-k}= & \nu+2^{2 / 3} a_k \nu^{1 / 3}+\frac{1}{5} 2^{4 / 3} a_k^2 \nu^{-1 / 3}+\left(\frac{11}{35}-\alpha^2-\frac{12}{175} a_k^3\right) \nu^{-1} \\
& +\left(\frac{16}{1575} a_k+\frac{92}{7875} a_k^4\right) 2^{2 / 3} \nu^{-5 / 3}-\left(\frac{15152}{3031875} a_k^5+\frac{1088}{121275} a_k^2\right) 2^{1 / 3} \nu^{-7 / 3}+\mathcal{O}\left(\nu^{-3}\right),
\end{aligned}</math>

For <math>\alpha \in (-1, 0)</math>,<ref name=":3" /><math display="block">\begin{aligned}
x_1=\frac{\alpha+1}{n} & +\frac{n-1}{2}\left(\frac{\alpha+1}{n}\right)^2-\frac{n^2+3 n-4}{12}\left(\frac{\alpha+1}{n}\right)^3 \\
& +\frac{7 n^3+6 n^2+23 n-36}{144}\left(\frac{\alpha+1}{n}\right)^4 \\
& -\frac{293 n^4+210 n^3+235 n^2+990 n-1728}{8640}\left(\frac{\alpha+1}{n}\right)^5+\cdots
\end{aligned}</math>

==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.<ref>{{Cite book|title=Quantum Mechanics in Chemistry|last=Ratner, Schatz|first=Mark A., George C.|publisher=Prentice Hall|year=2001|location=0-13-895491-7| pages=90–91}}</ref>

[[Vibronic coupling|Vibronic transitions]] in the Franck-Condon approximation can also be described using Laguerre polynomials.<ref>{{Cite journal|last1=Jong|first1=Mathijs de|last2=Seijo|first2=Luis|last3=Meijerink|first3=Andries| last4=Rabouw |first4=Freddy T.| date=2015-06-24|title=Resolving the ambiguity in the relation between Stokes shift and Huang–Rhys parameter |url=https://pubs.rsc.org/en/content/articlelanding/2015/cp/c5cp02093j|journal=Physical Chemistry Chemical Physics|language=en| volume=17 |issue=26|pages=16959–16969|doi=10.1039/C5CP02093J|pmid=26062123|bibcode=2015PCCP...1716959D|hdl=1874/321453|s2cid=34490576 | issn=1463-9084|hdl-access=free}}</ref>

==Multiplication theorems==
[[Arthur Erdélyi|Erdélyi]] gives the following two [[multiplication theorem]]s <ref>C. Truesdell, "[http://www.pnas.org/cgi/reprint/36/12/752.pdf On the Addition and Multiplication Theorems for the Special Functions]", ''Proceedings of the National Academy of Sciences, Mathematics'', (1950) pp. 752–757.</ref>

<math display="block">\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}</math>

== Relation to Hermite polynomials ==
The generalized Laguerre polynomials are related to the [[Hermite polynomial]]s:
<math display="block">\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}</math>
where the {{math|''H''<sub>''n''</sub>(''x'')}} are the [[Hermite polynomial]]s based on the weighting function {{math|exp(−''x''<sup>2</sup>)}}, the so-called "physicist's version."

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

Applying the addition formula,<math display="block">(-1)^n 2^{2n} n! \, L^{\left(\frac{r}{2}-1\right)}_{n}\Bigl(z_1^2+\cdots+z_r^2\Bigr)
=\sum_{m_1+\cdots+m_r=n} \prod_{i=1}^r H_{2m_i}(z_i).</math>

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

== Hardy–Hille formula ==
The generalized Laguerre polynomials satisfy the [[G. H. Hardy|Hardy]]–[[Einar Hille|Hille]] formula<ref>Szegő, p. 102.</ref><ref>{{Cite journal |last=Al-Salam |first=W. A. |date=1964-03-01 |title=Operational representations for the Laguerre and other polynomials |url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-31/issue-1/Operational-representations-for-the-Laguerre-and-other/10.1215/S0012-7094-64-03113-8.full |journal=Duke Mathematical Journal |volume=31 |issue=1 |doi=10.1215/S0012-7094-64-03113-8 |issn=0012-7094|url-access=subscription }}</ref>
<math display="block">\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),</math>
where the series on the left converges for <math>\alpha>-1</math> and <math>|t|<1</math>. Using the identity
<math display="block">\,_0F_1(;\alpha + 1;z)=\,\Gamma(\alpha + 1) z^{-\alpha/2} I_\alpha\left(2\sqrt{z}\right),</math>
(see [[Generalized hypergeometric function#The series 0F1|generalized hypergeometric function]]), this can also be written as
<math display="block">\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).</math>where <math>I_\alpha</math> denotes the modified Bessel function of the first kind, defined as<math display="block">
I_\alpha(z) = \sum_{k=0}^\infty \frac{1}{k!\, \Gamma(k+\alpha+1)} \left(\frac{z}{2}\right)^{2k+\alpha}
</math>This formula is a generalization of the [[Mehler kernel]] for [[Hermite polynomial]]s, which can be recovered from it by setting the Hermite polynomials as a special case of the associated Laguerre polynomials.

Substitute <math>t \mapsto -t/y</math> and take the <math>y \to \infty</math> limit, we obtain <ref>Szegő, page 102, Equation (5.1.16)</ref><math display="block">
\sum_{n=0}^\infty
\frac{t^n}{\Gamma(n+1+\alpha)} L_n^{(\alpha)}(x)
=
\frac{e^t}{(-xt)^{\alpha/2}}I_{\alpha}(2\sqrt{-xt}).</math>The formula is named after [[G. H. Hardy]] and [[Einar Hille]].<ref>G. H. Hardy, “Summation of a series of polynomials of Laguerre,” J. London Math. Soc., v. 7, 1932, pp. 138–139; addendum, 192.</ref><ref>E. Hille, “On Laguerre’s series. I, II, III,” Proc. Nat. Acad. Sci. U.S.A., v. 12, 1926, pp. 261–269; 348–352.</ref>

== Physics convention ==

The generalized Laguerre polynomials are used to describe the quantum wavefunction for [[hydrogen atom]] orbitals.<ref>{{cite book |last1=Griffiths |first1=David J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |location=Upper Saddle River, NJ |isbn=0131118927 |edition=2nd}}</ref><ref>{{cite book |last1=Sakurai |first1=J. J. |title=Modern quantum mechanics |date=2011 |publisher=Addison-Wesley |location=Boston |isbn=978-0805382914 |edition=2nd}}</ref><ref name="Merzbacher">{{cite book |last1=Merzbacher |first1=Eugen |title=Quantum mechanics |date=1998 |publisher=Wiley |location=New York |isbn=0471887021 |edition=3rd}}</ref> The convention used throughout this article expresses the generalized Laguerre polynomials as <ref>{{cite book |last1=Abramowitz |first1=Milton |title=Handbook of mathematical functions, with formulas, graphs, and mathematical tables |date=1965 |publisher=Dover Publications |location=New York |isbn=978-0-486-61272-0}}</ref>

<math display="block">L_n^{(\alpha)}(x) = \frac{\Gamma(\alpha + n + 1)}{\Gamma(\alpha + 1) n!} \,_1F_1(-n; \alpha + 1; x),</math>

where <math>\,_1F_1(a;b;x)</math> is the [[confluent hypergeometric function]].
In the physics literature,<ref name="Merzbacher" /> the generalized Laguerre polynomials are instead defined as

<math display="block">\bar{L}_n^{(\alpha)}(x) = \frac{\left[\Gamma(\alpha + n + 1)\right]^2}{\Gamma(\alpha + 1)n!} \,_1F_1(-n; \alpha + 1; x).</math>

The physics version is related to the standard version by

<math display="block">\bar{L}_n^{(\alpha)}(x) = (n+\alpha)! L_n^{(\alpha)}(x).</math>

There is yet another, albeit less frequently used, convention in the physics literature <ref>{{cite book |last1=Schiff |first1=Leonard I. |title=Quantum mechanics |date=1968 |publisher=McGraw-Hill |location=New York |isbn=0070856435 |edition=3d}}</ref><ref>{{cite book |last1=Messiah |first1=Albert |title=Quantum Mechanics. |date=2014 |publisher=Dover Publications |isbn=9780486784557}}</ref><ref>{{cite book |last1=Boas |first1=Mary L. |title=Mathematical methods in the physical sciences |date=2006 |publisher=Wiley |location=Hoboken, NJ |isbn=9780471198260 |edition=3rd}}</ref>

<math display="block">\tilde{L}_n^{(\alpha)}(x) = (-1)^{\alpha}\bar{L}_{n-\alpha}^{(\alpha)}.</math>

== Umbral calculus convention ==
Generalized Laguerre polynomials are linked to [[Umbral calculus]] by being [[Sheffer sequence]]s for <math>D/(D-I)</math> when multiplied by <math>n!</math>. In Umbral Calculus convention,<ref>{{Cite journal |last1=Rota |first1=Gian-Carlo |last2=Kahaner |first2=D |last3=Odlyzko |first3=A |date=1973-06-01 |title=On the foundations of combinatorial theory. VIII. Finite operator calculus |journal=Journal of Mathematical Analysis and Applications |language=en |volume=42 |issue=3 |pages=684–760 |doi=10.1016/0022-247X(73)90172-8 |issn=0022-247X|doi-access=free }}</ref> the default Laguerre polynomials are defined to be<math display="block">\mathcal L_n(x) = n!L_n^{(-1)}(x) = \sum_{k=0}^n L(n,k) (-x)^k</math>where <math display="inline">L(n,k) = \binom{n-1}{k-1} \frac{n!}{k!}</math> are the signless [[Lah number]]s. <math display="inline">(\mathcal L_n(x))_{n\in\N}</math> is a sequence of polynomials of [[binomial type]], ''ie'' they satisfy<math display="block">\mathcal L_n(x+y) = \sum_{k=0}^n \binom{n}{k} \mathcal L_k(x) \mathcal L_{n-k}(y)</math>

== See also ==
* [[Orthogonal polynomials]]
* [[Rodrigues' formula]]
* [[Angelescu polynomials]]
* [[Bessel polynomials]]
* [[Denisyuk polynomials]]
* [[Transverse mode]], an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile.

== Notes ==
{{Reflist|35em}}

== References ==
* {{Abramowitz_Stegun_ref|22|773}}
* {{cite book |last=Szegő |first=Gábor |author-link=Gábor Szegő |title=Orthogonal Polynomials |date=1975 |publisher=American Mathematical Society |edition=4th |series=American Mathematical Society Colloquium Publications |volume=23 |location=Providence, RI}}
* {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H.|last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof | last3=Koekoek ||first4=René F. |last4=Swarttouw}}
* {{cite book |last1=Spain |first1=B. |title=Functions of mathematical physics |last2=Smith |first2=M.G. |date=1970 |publisher=Van Nostrand Reinhold Company |location=London |chapter=10. Laguerre polynomials}}
* {{springer|title=Laguerre polynomials|id=p/l057310}}
* {{cite book|author=[[George Arfken]] and Hans Weber| title= Mathematical Methods for Physicists| publisher=Academic Press| year=2000| isbn = 978-0-12-059825-0 }}

== External links ==
* {{cite web|author=Timothy Jones|url=http://www.physics.drexel.edu/~tim/open/hydrofin | title=The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the Hydrogen Atom}}
* {{MathWorld|title=Laguerre polynomial|id=LaguerrePolynomial}}

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[[Category:Polynomials]]
[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]