Editing 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''' 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 + 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>}};
* 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>
<!-- \frac{n^\alpha}{\Gamma(\alpha+1)} + O\left(n^{\alpha-1}\right);</math> -->
* If {{math|''α''}} is non-negative, then ''L''<sub>''n''</sub><sup>(''α'')</sup> has ''n'' [[real number|real]], strictly positive [[Root of a function|roots]] (notice that <math>\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n</math> is a [[Sturm chain]]), which are all in the [[Interval (mathematics)|interval]] <math>\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \, \right].</math>{{citation needed|date=September 2011}}
* The polynomials' asymptotic behaviour for large {{mvar|n}}, but fixed {{mvar|α}} and {{math|''x'' > 0}}, is given by<ref>Szegő, p. 198.</ref><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> <math display="block">
\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}
</math> and summarizing by <math display="block">\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},</math> where <math>J_\alpha</math> is the [[Bessel function#Asymptotic forms|Bessel function]].

=== 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>

==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 Hardy–Hille formula<ref>Szegő, p. 102.</ref><ref>W. A. Al-Salam (1964), [https://projecteuclid.org/euclid.dmj/1077375084 "Operational representations for Laguerre and other polynomials"], ''Duke Math J.'' '''31''' (1): 127–142.</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>
== 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}}
* G. Szegő, ''Orthogonal polynomials'', 4th edition, ''Amer. Math. Soc. Colloq. Publ.'', vol. 23, Amer. Math. Soc., Providence, RI, 1975.
* {{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}}
* B. Spain, M.G. Smith, ''Functions of mathematical physics'', Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
* {{springer|title=Laguerre polynomials|id=p/l057310}}
* [[Eric W. Weisstein]], "[http://mathworld.wolfram.com/LaguerrePolynomial.html Laguerre Polynomial]", From MathWorld—A Wolfram Web Resource.
* {{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]]