Laguerre polynomials

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

Complex color plot of L_−1/9(z⁴) from −2−2i to 2+2i

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

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

The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form $${\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.}$$

These polynomials, usually denoted L₀, L₁, ..., are a polynomial sequence which may be defined by the Rodrigues formula,

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

They are orthogonal polynomials with respect to an inner product $${\displaystyle \langle f,g\rangle =\int _{0}^{\infty }f(x)g(x)e^{-x}\,dx.}$$

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

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

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

Recursive definition, closed form, and generating function

One can also define the Laguerre polynomials recursively, defining the first two polynomials as $${\displaystyle L_{0}(x)=1}$$ $${\displaystyle L_{1}(x)=1-x}$$ and then using the following recurrence relation for any k ≥ 1: $${\displaystyle L_{k+1}(x)={\frac {(2k+1-x)L_{k}(x)-kL_{k-1}(x)}{k+1}}.}$$ Furthermore, $${\displaystyle xL'_{n}(x)=nL_{n}(x)-nL_{n-1}(x).}$$

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

The closed form is $${\displaystyle L_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k}.}$$

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

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

A table of the Laguerre polynomials

n	${\displaystyle L_{n}(x)\,}$
0	${\displaystyle 1\,}$
1	${\displaystyle -x+1\,}$
2	${\displaystyle {\tfrac {1}{2}}(x^{2}-4x+2)\,}$
3	${\displaystyle {\tfrac {1}{6}}(-x^{3}+9x^{2}-18x+6)\,}$
4	${\displaystyle {\tfrac {1}{24}}(x^{4}-16x^{3}+72x^{2}-96x+24)\,}$
5	${\displaystyle {\tfrac {1}{120}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)\,}$
6	${\displaystyle {\tfrac {1}{720}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)\,}$
7	${\displaystyle {\tfrac {1}{5040}}(-x^{7}+49x^{6}-882x^{5}+7350x^{4}-29400x^{3}+52920x^{2}-35280x+5040)\,}$
8	${\displaystyle {\tfrac {1}{40320}}(x^{8}-64x^{7}+1568x^{6}-18816x^{5}+117600x^{4}-376320x^{3}+564480x^{2}-322560x+40320)\,}$
9	${\displaystyle {\tfrac {1}{362880}}(-x^{9}+81x^{8}-2592x^{7}+42336x^{6}-381024x^{5}+1905120x^{4}-5080320x^{3}+6531840x^{2}-3265920x+362880)\,}$
10	${\displaystyle {\tfrac {1}{3628800}}(x^{10}-100x^{9}+4050x^{8}-86400x^{7}+1058400x^{6}-7620480x^{5}[phone]x^{4}-72576000x^{3}[phone]x^{2}-36288000x+3628800)\,}$
n	${\displaystyle {\tfrac {1}{n!}}((-x)^{n}+n^{2}(-x)^{n-1}+\dots +n({n!})(-x)+n!)\,}$

The first six Laguerre polynomials.

Generalized Laguerre polynomials

For arbitrary real α the polynomial solutions of the differential equation^[2] $${\displaystyle x\,y''+\left(\alpha +1-x\right)y'+n\,y=0}$$ are called generalized Laguerre polynomials, or associated Laguerre polynomials.

One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as $${\displaystyle L_{0}^{(\alpha )}(x)=1}$$ $${\displaystyle L_{1}^{(\alpha )}(x)=1+\alpha -x}$$

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

The simple Laguerre polynomials are the special case α = 0 of the generalized Laguerre polynomials: $${\displaystyle L_{n}^{(0)}(x)=L_{n}(x).}$$

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

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

The first few generalized Laguerre polynomials, L_n^(k)(x)

Properties

• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as^[3] $${\displaystyle L_{n}^{(\alpha )}(x):={n+\alpha \choose n}M(-n,\alpha +1,x).}$$ where ${\textstyle {n+\alpha \choose n}}$ is a generalized binomial coefficient. When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression^[4] $${\displaystyle L_{n}^{(\alpha )}(x)={\frac {(-1)^{n}}{n!}}U(-n,\alpha +1,x)}$$ in terms of Kummer's function of the second kind.
• The closed form for these generalized Laguerre polynomials of degree n is^[5] $${\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}(-1)^{i}{n+\alpha \choose n-i}{\frac {x^{i}}{i!}}}$$ derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
• Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let ${\displaystyle D={\frac {d}{dx}}}$ and consider the differential operator ${\displaystyle M=xD^{2}+(\alpha +1)D}$. Then ${\displaystyle \exp(-tM)x^{n}=(-1)^{n}t^{n}n!L_{n}^{(\alpha )}\left({\frac {x}{t}}\right)}$.^{[citation needed]}
• The first few generalized Laguerre polynomials are:

n	${\displaystyle L_{n}^{(\alpha )}(x)\,}$
0	${\displaystyle 1\,}$
1	${\displaystyle -x+\alpha +1\,}$
2	${\displaystyle {\tfrac {1}{2}}(x^{2}-2\left(\alpha +2\right)x+\left(\alpha +1\right)\left(\alpha +2\right))\,}$
3	${\displaystyle {\tfrac {1}{6}}(-x^{3}+3\left(\alpha +3\right)x^{2}-3\left(\alpha +2\right)\left(\alpha +3\right)x+\left(\alpha +1\right)\left(\alpha +2\right)\left(\alpha +3\right))\,}$
4	${\displaystyle {\tfrac {1}{24}}(x^{4}-4\left(\alpha +4\right)x^{3}+6\left(\alpha +3\right)\left(\alpha +4\right)x^{2}-4\left(\alpha +2\right)\cdots \left(\alpha +4\right)x+\left(\alpha +1\right)\cdots \left(\alpha +4\right))\,}$
5	${\displaystyle {\tfrac {1}{120}}(-x^{5}+5\left(\alpha +5\right)x^{4}-10\left(\alpha +4\right)\left(\alpha +5\right)x^{3}+10\left(\alpha +3\right)\cdots \left(\alpha +5\right)x^{2}-5\left(\alpha +2\right)\cdots \left(\alpha +5\right)x+\left(\alpha +1\right)\cdots \left(\alpha +5\right))\,}$
6	${\displaystyle {\tfrac {1}{720}}(x^{6}-6\left(\alpha +6\right)x^{5}+15\left(\alpha +5\right)\left(\alpha +6\right)x^{4}-20\left(\alpha +4\right)\cdots \left(\alpha +6\right)x^{3}+15\left(\alpha +3\right)\cdots \left(\alpha +6\right)x^{2}-6\left(\alpha +2\right)\cdots \left(\alpha +6\right)x+\left(\alpha +1\right)\cdots \left(\alpha +6\right))\,}$
7	${\displaystyle {\tfrac {1}{5040}}(-x^{7}+7\left(\alpha +7\right)x^{6}-21\left(\alpha +6\right)\left(\alpha +7\right)x^{5}+35\left(\alpha +5\right)\cdots \left(\alpha +7\right)x^{4}-35\left(\alpha +4\right)\cdots \left(\alpha +7\right)x^{3}+21\left(\alpha +3\right)\cdots \left(\alpha +7\right)x^{2}-7\left(\alpha +2\right)\cdots \left(\alpha +7\right)x+\left(\alpha +1\right)\cdots \left(\alpha +7\right))\,}$
8	${\displaystyle {\tfrac {1}{40320}}(x^{8}-8\left(\alpha +8\right)x^{7}+28\left(\alpha +7\right)\left(\alpha +8\right)x^{6}-56\left(\alpha +6\right)\cdots \left(\alpha +8\right)x^{5}+70\left(\alpha +5\right)\cdots \left(\alpha +8\right)x^{4}-56\left(\alpha +4\right)\cdots \left(\alpha +8\right)x^{3}+28\left(\alpha +3\right)\cdots \left(\alpha +8\right)x^{2}-8\left(\alpha +2\right)\cdots \left(\alpha +8\right)x+\left(\alpha +1\right)\cdots \left(\alpha +8\right))\,}$
9	${\displaystyle {\tfrac {1}{362880}}(-x^{9}+9\left(\alpha +9\right)x^{8}-36\left(\alpha +8\right)\left(\alpha +9\right)x^{7}+84\left(\alpha +7\right)\cdots \left(\alpha +9\right)x^{6}-126\left(\alpha +6\right)\cdots \left(\alpha +9\right)x^{5}+126\left(\alpha +5\right)\cdots \left(\alpha +9\right)x^{4}-84\left(\alpha +4\right)\cdots \left(\alpha +9\right)x^{3}+36\left(\alpha +3\right)\cdots \left(\alpha +9\right)x^{2}-9\left(\alpha +2\right)\cdots \left(\alpha +9\right)x+\left(\alpha +1\right)\cdots \left(\alpha +9\right))\,}$
10	${\displaystyle {\tfrac {1}{3628800}}(x^{10}-10\left(\alpha +10\right)x^{9}+45\left(\alpha +9\right)\left(\alpha +10\right)x^{8}-120\left(\alpha +8\right)\cdots \left(\alpha +10\right)x^{7}+210\left(\alpha +7\right)\cdots \left(\alpha +10\right)x^{6}-252\left(\alpha +6\right)\cdots \left(\alpha +10\right)x^{5}+210\left(\alpha +5\right)\cdots \left(\alpha +10\right)x^{4}-120\left(\alpha +4\right)\cdots \left(\alpha +10\right)x^{3}+45\left(\alpha +3\right)\cdots \left(\alpha +10\right)x^{2}-10\left(\alpha +2\right)\cdots \left(\alpha +10\right)x+\left(\alpha +1\right)\cdots \left(\alpha +10\right))\,}$

• The coefficient of the leading term is (−1)ⁿ/n!;
• The constant term, which is the value at 0, is $${\displaystyle L_{n}^{(\alpha )}(0)={n+\alpha \choose n}={\frac {\Gamma (n+\alpha +1)}{n!\,\Gamma (\alpha +1)}};}$$
• The discriminant is^[6]$${\displaystyle \operatorname {Disc} \left(L_{n}^{(\alpha )}\right)=\prod _{j=1}^{n}j^{j-2n+2}(j+\alpha )^{j-1}}$$

As a contour integral

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

Recurrence relations

The addition formula for Laguerre polynomials:^[7] $${\displaystyle L_{n}^{(\alpha _{1}+\dots +\alpha _{r}+r-1)}\left(x_{1}+\dots +x_{r}\right)=\sum _{m_{1}+\dots +m_{r}=n}L_{m_{1}}^{(\alpha _{1})}\left(x_{1}\right)\cdots L_{m_{r}}^{(\alpha _{r})}\left(x_{r}\right).}$$Laguerre's polynomials satisfy the recurrence relations $${\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}L_{n-i}^{(\alpha +i)}(y){\frac {(y-x)^{i}}{i!}},}$$ in particular $${\displaystyle L_{n}^{(\alpha +1)}(x)=\sum _{i=0}^{n}L_{i}^{(\alpha )}(x)}$$ and $${\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n-i-1 \choose n-i}L_{i}^{(\beta )}(x),}$$ or $${\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n \choose n-i}L_{i}^{(\beta -i)}(x);}$$ moreover $${\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)-\sum _{j=0}^{\Delta -1}{n+\alpha \choose n-j}(-1)^{j}{\frac {x^{j}}{j!}}&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(\alpha +\Delta )}(x)\\[6pt]&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha -i-1 \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(n+\alpha +\Delta -i)}(x)\end{aligned}}}$$

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

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

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

Derivatives

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

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

Moreover, the following equation holds: $${\displaystyle {\frac {1}{k!}}{\frac {d^{k}}{dx^{k}}}x^{\alpha }L_{n}^{(\alpha )}(x)={n+\alpha \choose k}x^{\alpha -k}L_{n}^{(\alpha -k)}(x),}$$ which generalizes with Cauchy's formula to $${\displaystyle L_{n}^{(\alpha ')}(x)=(\alpha '-\alpha ){\alpha '+n \choose \alpha '-\alpha }\int _{0}^{x}{\frac {t^{\alpha }(x-t)^{\alpha '-\alpha -1}}{x^{\alpha '}}}L_{n}^{(\alpha )}(t)\,dt.}$$

The derivative with respect to the second variable α has the form,^[8] $${\displaystyle {\frac {d}{d\alpha }}L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n-1}{\frac {L_{i}^{(\alpha )}(x)}{n-i}}.}$$ The generalized Laguerre polynomials obey the differential equation $${\displaystyle xL_{n}^{(\alpha )\prime \prime }(x)+(\alpha +1-x)L_{n}^{(\alpha )\prime }(x)+nL_{n}^{(\alpha )}(x)=0,}$$ which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

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

In Sturm–Liouville form the differential equation is

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

which shows that L^(α)
_n is an eigenvector for the eigenvalue n.

Orthogonality

The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function x^α e^−x:^[9]

$${\displaystyle \int _{0}^{\infty }x^{\alpha }e^{-x}L_{n}^{(\alpha )}(x)L_{m}^{(\alpha )}(x)dx={\frac {\Gamma (n+\alpha +1)}{n!}}\delta _{n,m},}$$

which follows from

$${\displaystyle \int _{0}^{\infty }x^{\alpha '-1}e^{-x}L_{n}^{(\alpha )}(x)dx={\alpha -\alpha '+n \choose n}\Gamma (\alpha ').}$$

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

$${\displaystyle \int _{0}^{\infty }L_{n}^{(\alpha )}(x)L_{m}^{(\alpha )}(x)\Gamma (x,\alpha +1,1)dx={n+\alpha \choose n}\delta _{n,m}.}$$

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)^{[citation needed]}

$${\displaystyle {\begin{aligned}K_{n}^{(\alpha )}(x,y)&:={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {L_{i}^{(\alpha )}(x)L_{i}^{(\alpha )}(y)}{\alpha +i \choose i}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}{\frac {L_{n}^{(\alpha )}(x)L_{n+1}^{(\alpha )}(y)-L_{n+1}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)}{{\frac {x-y}{n+1}}{n+\alpha \choose n}}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {x^{i}}{i!}}{\frac {L_{n-i}^{(\alpha +i)}(x)L_{n-i}^{(\alpha +i+1)}(y)}{{\alpha +n \choose n}{n \choose i}}};\end{aligned}}}$$

recursively

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

Moreover,^{[clarification needed Limit as n goes to infinity?]}

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

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

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

$${\displaystyle \int _{0}^{\infty }x^{\alpha +1}e^{-x}\left[L_{n}^{(\alpha )}(x)\right]^{2}dx={\frac {(n+\alpha )!}{n!}}(2n+\alpha +1).}$$

Series expansions

Let a function have the (formal) series expansion $${\displaystyle f(x)=\sum _{i=0}^{\infty }f_{i}^{(\alpha )}L_{i}^{(\alpha )}(x).}$$

Then $${\displaystyle f_{i}^{(\alpha )}=\int _{0}^{\infty }{\frac {L_{i}^{(\alpha )}(x)}{i+\alpha \choose i}}\cdot {\frac {x^{\alpha }e^{-x}}{\Gamma (\alpha +1)}}\cdot f(x)\,dx.}$$

The series converges in the associated Hilbert space L²[0, ∞) if and only if

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

Further examples of expansions

Monomials are represented as $${\displaystyle {\frac {x^{n}}{n!}}=\sum _{i=0}^{n}(-1)^{i}{n+\alpha \choose n-i}L_{i}^{(\alpha )}(x),}$$ while binomials have the parametrization $${\displaystyle {n+x \choose n}=\sum _{i=0}^{n}{\frac {\alpha ^{i}}{i!}}L_{n-i}^{(x+i)}(\alpha ).}$$

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

Asymptotics

In terms of elementary functions

For any fixed positive integer ${\displaystyle M}$, fixed real number ${\displaystyle \alpha }$, fixed and bounded interval ${\displaystyle [c,d]\subset (0,+\infty )}$, uniformly for ${\displaystyle x\in [c,d]}$, at ${\displaystyle n\to \infty }$:$${\displaystyle L_{n}^{(\alpha )}\left(x\right)={\frac {n^{{\frac {1}{2}}\alpha -{\frac {1}{4}}}{\mathrm {e} }^{{\frac {1}{2}}x}}{{\pi }^{\frac {1}{2}}x^{{\frac {1}{2}}\alpha +{\frac {1}{4}}}}}\left(\cos \theta _{n}^{(\alpha )}(x)\left(\sum _{m=0}^{M-1}{\frac {a_{m}(x)}{n^{{\frac {1}{2}}m}}}+O\left({\frac {1}{n^{{\frac {1}{2}}M}}}\right)\right)+\sin \theta _{n}^{(\alpha )}(x)\left(\sum _{m=1}^{M-1}{\frac {b_{m}(x)}{n^{{\frac {1}{2}}m}}}+O\left({\frac {1}{n^{{\frac {1}{2}}M}}}\right)\right)\right)}$$where $${\displaystyle \theta _{n}^{(\alpha )}(x):=2(nx)^{\frac {1}{2}}-\left({\tfrac {1}{2}}\alpha +{\tfrac {1}{4}}\right)\pi .}$$and ${\displaystyle a_{0},b_{1},a_{1},b_{2},\dots }$ are functions depending on ${\displaystyle \alpha ,x}$ but not ${\displaystyle n}$, and regular for ${\displaystyle x>0}$. The first few ones are:$${\displaystyle {\begin{aligned}&a_{0}(x)=1\\&a_{1}(x)=0\\&b_{1}(x)={\frac {1}{48x^{\frac {1}{2}}}}\left(4x^{2}-24(\alpha +1)x+3-12\alpha ^{2}\right)\end{aligned}}}$$This is Perron's formula.^{[10][11]: 78} There is also a generalization for ${\displaystyle x\in \mathbb {C} \setminus [0,\infty )}$.^[12] Fejér's formula is a special case of Perron's formula with ${\displaystyle M=1}$.^[13][12][14]

In terms of Bessel functions

The Mehler–Heine formula states:

${\displaystyle \lim _{n\to \infty }n^{-\alpha }L_{n}^{(\alpha )}\left({\frac {z^{2}}{4n}}\right)=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z),}$

where ${\displaystyle J_{\alpha }}$ is a Bessel function of the first kind.

See also: ^[10].

In terms of Airy functions

Let ${\displaystyle \nu =4n+2\alpha +2}$. Let ${\displaystyle \operatorname {Ai} }$ be the Airy function. Let ${\displaystyle \alpha }$ be arbitrary and real, ${\displaystyle \epsilon }$ and ${\displaystyle \omega }$ be positive and fixed.

The Plancherel–Rotach asymptotics formulas:^[15][10]

• for ${\displaystyle x=\nu \cos ^{2}\varphi }$ and ${\displaystyle \epsilon \leq \varphi \leq {\tfrac {\pi }{2}}-\epsilon n^{-1/2}}$

${\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}(\pi \sin \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}{\big \{}\sin \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(\sin 2\varphi -2\varphi )+3\pi /4\right]+(nx)^{-1/2}{\mathcal {O}}(1){\big \}}}$

• for ${\displaystyle x=\nu \cosh ^{2}\varphi }$ and ${\displaystyle \epsilon \leq \varphi \leq \omega }$

${\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)={\tfrac {1}{2}}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}\exp \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(2\varphi -\sinh 2\varphi )\right]\{1+{\mathcal {O}}\left(n^{-1}\right)\}}$

• for ${\displaystyle x=\nu -2(2n/3)^{1/3}t}$ and ${\displaystyle t}$ complex and bounded

${\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}\pi ^{-1}2^{-\alpha -1/3}3^{1/3}n^{-1/3}{\bigg \{}\pi \operatorname {Ai} (-3^{-1/3}t)+{\mathcal {O}}\left(n^{-2/3}\right){\bigg \}}}$

See also: ^[10].

Zeroes

Notation

${\displaystyle j_{\alpha ,m}}$ is the ${\displaystyle m}$-th positive zero of the Bessel function ${\displaystyle J_{\alpha }(x)}$.

${\displaystyle a_{m}}$ is the ${\displaystyle m}$-th zero of the Airy function ${\displaystyle \operatorname {Ai} (x)}$, in descending order: ${\displaystyle 0>a_{1}>a_{2}>\cdots }$.

${\displaystyle \nu =4n+2\alpha +2}$.

If ${\displaystyle \alpha >-1}$, then ${\displaystyle L_{n}^{(\alpha )}}$ has ${\displaystyle n}$ real roots. Thus in this section we assume ${\displaystyle \alpha >-1}$ by default.

${\displaystyle x_{1}<\dots <x_{n}}$ are the real roots of ${\displaystyle L_{n}^{(\alpha )}}$.

Note that ${\displaystyle \left((-1)^{n-i}L_{n-i}^{(\alpha )}\right)_{i=0}^{n}}$ is a Sturm chain.

Inequalities

For ${\displaystyle \alpha >-1}$, we have these bounds:^{[16][17][6][18]}

• ${\displaystyle x_{1}<{\frac {(\alpha +1)(\alpha +2)}{n+\alpha +1}}}$
• ${\displaystyle x_{1}<{\frac {(\alpha +1)(\alpha +3)}{2n+\alpha +1}}}$
• ${\displaystyle x_{1}<{\frac {(\alpha +1)(\alpha +2)(\alpha +4)(2n+\alpha +1)}{(\alpha +1)^{2}(\alpha +2)+n(5\alpha +11)(n+\alpha +1)}}}$
• ${\displaystyle x_{n}\leq 2n+\alpha -1+2{\sqrt {(n-2)(n+\alpha -1)}}}$ when ${\displaystyle n\geq 2}$
• ${\displaystyle x_{n}>4n+\alpha -16{\sqrt {2n}}}$
• ${\displaystyle x_{n}>3n-4}$
• ${\displaystyle x_{n}>2n+\alpha -1}$
• ${\displaystyle x_{n}>2n+\alpha -2+{\sqrt {n^{2}-2n+\alpha n+2}}}$
• $${\displaystyle {\begin{aligned}&(n+2)x_{1}&\geq \left(n-1-{\sqrt {n^{2}+(n+2)(\alpha +1)}}\right)^{2}-1\\&(n+2)x_{n}&\leq \left(n-1+{\sqrt {n^{2}+(n+2)(\alpha +1)}}\right)^{2}-1\end{aligned}}}$$
• $${\displaystyle {\begin{aligned}x_{1}&>{\frac {1}{2}}\nu -3-{\sqrt {1+4(n-1)(n+\alpha -1)}}\\x_{n}&<{\frac {1}{2}}\nu -3+{\sqrt {1+4(n-1)(n+\alpha -1)}}\end{aligned}}}$$

For fixed ${\displaystyle k=1,\dots ,n}$,^[16][6][17]$${\displaystyle {\begin{aligned}\nu x_{k}&>j_{\alpha ,k}^{2}\\x_{k}&<{\frac {j_{\alpha ,k}^{2}}{\nu /2+{\sqrt {(\nu /2)^{2}-j_{\alpha ,k}^{2}}}}}\quad {\text{ if }}\nu /2>j_{\alpha ,k}\\x_{k}&<\left[\nu ^{1/2}+2^{-1/3}\nu ^{-1/6}a_{n-k+1}\right]^{2}\quad {\text{ if }}|\alpha |\geqslant 1/4\\x_{k}&<\nu +2^{\frac {2}{3}}a_{k}\nu ^{\frac {1}{3}}+2^{-{\frac {2}{3}}}a_{k}^{2}\nu ^{-{\frac {1}{3}}}\end{aligned}}}$$For fixed ${\displaystyle k}$, we have ${\displaystyle \lim _{n\to \infty }\nu x_{k}=j_{\alpha ,k}^{2}}$, so the first inequality is sharp.

See also ^[19].

Electrostatics

The zeroes satisfy the Stieltjes relations:^[20][21]$${\displaystyle {\begin{aligned}\sum _{1\leq j\leq n,i\neq j}{\frac {1}{x_{i}-x_{j}}}&={\frac {1}{2}}\left(1-{\frac {\alpha +1}{x_{i}}}\right)\\\sum _{1\leq j\leq n}{\frac {1}{x_{j}}}&={\frac {n}{\alpha +1}}\\\sum _{1\leq j\leq n,i\neq j}{\frac {1}{(x_{i}-x_{j})^{2}}}&=-{\frac {(\alpha +1)(\alpha +5)}{12x_{i}^{2}}}+{\frac {2n+\alpha +1}{6x_{i}}}-{\frac {1}{12}}\\\sum _{1\leq j\leq n,i\neq j}{\frac {1}{(x_{i}-x_{j})^{3}}}&=-{\frac {(\alpha +1)(\alpha +3)}{8x_{i}^{3}}}+{\frac {2n+\alpha +1}{8x_{i}^{2}}}\\\end{aligned}}}$$The first relation can be interpreted physically. Fix an electric particle at origin with charge ${\displaystyle +{\frac {\alpha +1}{2}}}$, and produce a constant electric field of strength ${\displaystyle -{\frac {1}{2}}}$. Then, place ${\displaystyle n}$ electric particles with charge ${\displaystyle +1}$. The first relation states that the zeroes of ${\displaystyle L_{n}^{(\alpha )}}$ 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^[22]$${\displaystyle \sum _{i=1}^{n}{\frac {1}{x-x_{i}}}=-\sum _{k=0}^{\infty }S_{k+1}x^{k},\quad S_{k}:=\sum _{i=1}^{n}x_{i}^{-k}}$$which allows the following bound$${\displaystyle S_{m}^{-1/m}<x_{1}<S_{m}/S_{m+1},\quad m=1,2,\ldots }$$

Limit distribution

Let ${\displaystyle F_{n}(t):={\frac {1}{n}}\#\{i:x_{i}\leq t\}}$ be the cumulative distribution function for the roots, then we have the limit law^[23]$${\displaystyle \lim _{n\to \infty }F_{n}(4nt)={\frac {2}{\pi }}\int _{0}^{t}{\sqrt {\frac {1-s}{s}}}ds\quad \forall t\in (0,1]}$$which can be interpreted as the limit distribution of the Wishart ensemble spectrum.

For fixed ${\displaystyle \alpha >-1}$ and fixed ${\displaystyle k}$, as ${\displaystyle n\to \infty }$,^[17]$${\displaystyle {\begin{aligned}x_{n+1-k}=&\nu +2^{2/3}a_{k}\nu ^{1/3}+{\frac {1}{5}}2^{4/3}a_{k}^{2}\nu ^{-1/3}+\left({\frac {11}{35}}-\alpha ^{2}-{\frac {12}{175}}a_{k}^{3}\right)\nu ^{-1}\\&+\left({\frac {16}{1575}}a_{k}+{\frac {92}{7875}}a_{k}^{4}\right)2^{2/3}\nu ^{-5/3}-\left({\frac {15152}{3031875}}a_{k}^{5}+{\frac {1088}{121275}}a_{k}^{2}\right)2^{1/3}\nu ^{-7/3}+{\mathcal {O}}\left(\nu ^{-3}\right),\end{aligned}}}$$

For ${\displaystyle \alpha \in (-1,0)}$,^[22]$${\displaystyle {\begin{aligned}x_{1}={\frac {\alpha +1}{n}}&+{\frac {n-1}{2}}\left({\frac {\alpha +1}{n}}\right)^{2}-{\frac {n^{2}+3n-4}{12}}\left({\frac {\alpha +1}{n}}\right)^{3}\\&+{\frac {7n^{3}+6n^{2}+23n-36}{144}}\left({\frac {\alpha +1}{n}}\right)^{4}\\&-{\frac {293n^{4}+210n^{3}+235n^{2}+990n-1728}{8640}}\left({\frac {\alpha +1}{n}}\right)^{5}+\cdots \end{aligned}}}$$

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.^[24]

Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.^[25]

Multiplication theorems

Erdélyi gives the following two multiplication theorems ^[26]

$${\displaystyle {\begin{aligned}&t^{n+1+\alpha }e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=n}^{\infty }{k \choose n}\left(1-{\frac {1}{t}}\right)^{k-n}L_{k}^{(\alpha )}(z),\\[6pt]&e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=0}^{\infty }{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{(\alpha +k)}(z).\end{aligned}}}$$

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials: $${\displaystyle {\begin{aligned}H_{2n}(x)&=(-1)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\\[4pt]H_{2n+1}(x)&=(-1)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\end{aligned}}}$$ where the H_n(x) are the Hermite polynomials based on the weighting function exp(−x²), 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,$${\displaystyle (-1)^{n}2^{2n}n!\,L_{n}^{\left({\frac {r}{2}}-1\right)}{\Bigl (}z_{1}^{2}+\cdots +z_{r}^{2}{\Bigr )}=\sum _{m_{1}+\cdots +m_{r}=n}\prod _{i=1}^{r}H_{2m_{i}}(z_{i}).}$$

Relation to hypergeometric functions

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

Hardy–Hille formula

The generalized Laguerre polynomials satisfy the Hardy–Hille formula^[27][28] $${\displaystyle \sum _{n=0}^{\infty }{\frac {n!\,\Gamma \left(\alpha +1\right)}{\Gamma \left(n+\alpha +1\right)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{(1-t)^{\alpha +1}}}e^{-(x+y)t/(1-t)}\,_{0}F_{1}\left(;\alpha +1;{\frac {xyt}{(1-t)^{2}}}\right),}$$ where the series on the left converges for ${\displaystyle \alpha >-1}$ and ${\displaystyle |t|<1}$. Using the identity $${\displaystyle \,_{0}F_{1}(;\alpha +1;z)=\,\Gamma (\alpha +1)z^{-\alpha /2}I_{\alpha }\left(2{\sqrt {z}}\right),}$$ (see generalized hypergeometric function), this can also be written as $${\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{\Gamma (1+\alpha +n)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{(xyt)^{\alpha /2}(1-t)}}e^{-(x+y)t/(1-t)}I_{\alpha }\left({\frac {2{\sqrt {xyt}}}{1-t}}\right).}$$where ${\displaystyle I_{\alpha }}$ denotes the modified Bessel function of the first kind, defined as$${\displaystyle I_{\alpha }(z)=\sum _{k=0}^{\infty }{\frac {1}{k!\,\Gamma (k+\alpha +1)}}\left({\frac {z}{2}}\right)^{2k+\alpha }}$$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 ${\displaystyle t\mapsto -t/y}$ and take the ${\displaystyle y\to \infty }$ limit, we obtain ^[29]$${\displaystyle \sum _{n=0}^{\infty }{\frac {t^{n}}{\Gamma (n+1+\alpha )}}L_{n}^{(\alpha )}(x)={\frac {e^{t}}{(-xt)^{\alpha /2}}}I_{\alpha }(2{\sqrt {-xt}}).}$$The formula is named after G. H. Hardy and Einar Hille.^[30][31]

Physics convention

The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals.^[32][33][34] The convention used throughout this article expresses the generalized Laguerre polynomials as ^[35]

$${\displaystyle L_{n}^{(\alpha )}(x)={\frac {\Gamma (\alpha +n+1)}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x),}$$

where ${\displaystyle \,_{1}F_{1}(a;b;x)}$ is the confluent hypergeometric function. In the physics literature,^[34] the generalized Laguerre polynomials are instead defined as

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

The physics version is related to the standard version by

$${\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)=(n+\alpha )!L_{n}^{(\alpha )}(x).}$$

There is yet another, albeit less frequently used, convention in the physics literature ^[36][37][38]

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

Umbral calculus convention

Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for ${\displaystyle D/(D-I)}$ when multiplied by ${\displaystyle n!}$. In Umbral Calculus convention,^[39] the default Laguerre polynomials are defined to be$${\displaystyle {\mathcal {L}}_{n}(x)=n!L_{n}^{(-1)}(x)=\sum _{k=0}^{n}L(n,k)(-x)^{k}}$$where ${\textstyle L(n,k)={\binom {n-1}{k-1}}{\frac {n!}{k!}}}$ are the signless Lah numbers. ${\textstyle ({\mathcal {L}}_{n}(x))_{n\in \mathbb {N} }}$ is a sequence of polynomials of binomial type, ie they satisfy$${\displaystyle {\mathcal {L}}_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}{\mathcal {L}}_{k}(x){\mathcal {L}}_{n-k}(y)}$$

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.