Laguerre polynomials
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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 solutions only if Template:Mvar 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 Template:Mvar 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>Template:Cite journal</ref> Nikolay Yakovlevich Sonin).
More generally, a Laguerre function is a solution when Template:Mvar 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 Template:Math, Template:Math, ..., are a polynomial sequence which may be defined by the 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 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.)
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 recurrence relation for any Template:Math: <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>
| n | <math>L_n(x)\,</math> |
|---|---|
| 0 | <math>1\,</math> |
| 1 | <math>-x+1\,</math> |
| 2 | <math> \tfrac{1}{2} (x^2-4x+2) \,</math> |
| 3 | <math>\tfrac{1}{6} (-x^3+9x^2-18x+6) \,</math> |
| 4 | <math>\tfrac{1}{24} (x^4-16x^3+72x^2-96x+24) \,</math> |
| 5 | <math>\tfrac{1}{120} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math> |
| 6 | <math>\tfrac{1}{720} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math> |
| 7 | <math>\tfrac{1}{5040} (-x^7+49x^6-882x^5+7350x^4-29400x^3+52920x^2-35280x+5040) \,</math> |
| 8 | <math>\tfrac{1}{40320} (x^8-64x^7+1568x^6-18816x^5+117600x^4-376320x^3+564480x^2-322560x+40320) \,</math> |
| 9 | <math>\tfrac{1}{362880} (-x^9+81x^8-2592x^7+42336x^6-381024x^5+1905120x^4-5080320x^3+6531840x^2-3265920x+362880) \,</math> |
| 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> |
| 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> |
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 recurrence relation for any Template:Math: <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 Template:Math 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>
Properties
- Laguerre functions are defined by confluent hypergeometric functions 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 Template:Mvar is an integer the function reduces to a polynomial of degree Template:Mvar. 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 Kummer's function of the second kind.
- The closed form for these generalized Laguerre polynomials of degree Template:Mvar 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'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>.Template:Citation needed
- The first few generalized Laguerre polynomials are:
| n | <math>L_n^{(\alpha)}(x)\,</math> |
|---|---|
| 0 | <math>1\,</math> |
| 1 | <math>-x+\alpha +1\,</math> |
| 2 | <math> \tfrac{1}{2} (x^2-2\left( \alpha +2 \right) x+\left( \alpha +1 \right) \left( \alpha +2 \right)) \,</math> |
| 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> |
| 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> |
| 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> |
| 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> |
| 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> |
| 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> |
| 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> |
| 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 Template:Math;
- The constant term, which is the value at 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">Template:Cite web</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>Template:Cite web</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} \fracTemplate: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} \fracTemplate: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 Template:Mvar 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 Template:Mvar 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 (Template:Math) of the formula above: for integer Template:Math 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 Template:Mvar 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 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 Template:Mvar has the form,<ref>Template:Cite journal</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 kth 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 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 Template:Math is an eigenvector for the eigenvalue Template:Mvar.
Orthogonality
The generalized Laguerre polynomials are orthogonal over Template:Closed-open with respect to the measure with weighting function Template:Math:<ref>Template:Cite web</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)Template:Citation needed
<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)}Template:\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)}Template:\alpha+n \choose n.</math>
Moreover,Template:Clarify
<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} \fracTemplate:\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 mechanical treatment of the 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)}Template: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 Template:Math 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
Monomials 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 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
Template:AnchorFor 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 Perron's formula.<ref name=":1">Template:Cite web</ref><ref>Template:Cite journal</ref>Template:Reference page There is also a generalization for <math>x \in \mathbb C \setminus [0, \infty) </math>.<ref name=":0">Szegő, p. 198.</ref> Fejér's formula is a special case of Perron's formula with <math>M = 1</math>.<ref>Template:Citation</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 {{#invoke:CS1 identifiers|main|_template=doi}}</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 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">Template:Cite journal</ref><ref name=":5">Template:Cite journal</ref><ref name=":2" /><ref>Template:Cite journal</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>Template:Harvard citation</ref>
Electrostatics
The zeroes satisfy the Stieltjes relations:<ref>Template:Cite journal</ref><ref>Template:Harvard citation</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">Template:Cite journal</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>Template:Cite journal</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>Template:Cite book</ref>
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.<ref>Template:Cite journal</ref>
Multiplication theorems
Erdélyi gives the following two multiplication theorems <ref>C. Truesdell, "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 polynomials: <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 Template:Math are the Hermite polynomials based on the weighting function Template:Math, 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 functions, specifically the confluent hypergeometric functions, 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>Template:Cite journal</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), 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 polynomials, 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>Template:Cite book</ref><ref>Template:Cite book</ref><ref name="Merzbacher">Template:Cite book</ref> The convention used throughout this article expresses the generalized Laguerre polynomials as <ref>Template:Cite book</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>Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</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 sequences for <math>D/(D-I)</math> when multiplied by <math>n!</math>. In Umbral Calculus convention,<ref>Template:Cite journal</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 numbers. <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
References
- Template:Abramowitz Stegun ref
- Template:Cite book
- Template:Dlmf
- Template:Cite book
- Template:Springer
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