Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886), are solutions of Laguerre's equation:

$x\,y'' + (1 - x)\,y' + n\,y = 0\,$

which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials (also named Sonin polynomials after Nikolay Yakovlevich Sonin in some older books) are solutions of

$x\,y'' + (\alpha+1 - x)\,y' + n\,y = 0\,$

The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form

$\int_0^\infty f(x) e^{-x} \, dx.$

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

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).$

They are orthogonal polynomials with respect to an inner product

$\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.$

The sequence of Laguerre polynomials is a Sheffer sequence.

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

The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.

Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of n!, than the definition used here. (Furthermore, various physicist use somewhat different definitions of the so-called associated Laguerre polynomials.)

The first few polynomials

These are the first few Laguerre polynomials:

n	$L_n(x)\,$
0	$1\,$
1	$-x+1\,$
2	${\scriptstyle\frac{1}{2}} (x^2-4x+2) \,$
3	${\scriptstyle\frac{1}{6}} (-x^3+9x^2-18x+6) \,$
4	${\scriptstyle\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \,$
5	${\scriptstyle\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \,$
6	${\scriptstyle\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,$

The first six Laguerre polynomials.

Recursive definition

We can also define the Laguerre polynomials recursively, defining the first two polynomials as

$L_0(x) = 1\,$

$L_1(x) = 1 - x\,$

and then using the following recurrence relation for any k ≥ 1:

$L_{k + 1}(x) = \frac{1}{k + 1} \left( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\right).$

The closed form is

$L_k(x)=\sum_{i=0}^k \binom{k}{i}\frac{(-1)^i}{i!}x^i.$

Generalized Laguerre polynomials

For arbitrary real α the polynomial solutions of the differential equation

$x\,y'' + (\alpha +1 - x)\,y' + n\,y = 0$

are called generalized Laguerre polynomials, or associated Laguerre polynomials. The Rodrigues' formula for them are

$L_n^{(\alpha)}(x)= {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right).$

The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:

$L^{(0)}_n(x)=L_n(x).$

Explicit examples and properties of generalized Laguerre polynomials

• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as

$L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).$

When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression

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

in terms of Kummer's function of the second kind.

• The generalized Laguerre polynomial of degree n is

$L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}$

(derived equivalently by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.)

• The first few generalized Laguerre polynomials are:

$\begin{align} L_0^{(\alpha)} (x) & = 1 \\ L_1^{(\alpha)}(x) & = -x + \alpha +1 \\ L_2^{(\alpha)}(x) & = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2} \\ L_3^{(\alpha)}(x) & = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2} + \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6} \end{align}$

• The coefficient of the leading term is (−1)ⁿ/n!;
• The constant term, which is the value at 0, is

$L_n^{(\alpha)}(0)= {n+\alpha\choose n} \approx \frac{n^\alpha}{\Gamma(\alpha+1)};$

• L_n^(α) has n real, strictly positive roots (notice that $\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n$ is a Sturm chain), which are all in the interval $\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \right].$^{[citation needed]}

• The polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by

$L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \cos\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha+\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right),$

$L_n^{(\alpha)}(-x) = \frac{(n+1)^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} e^{2 \sqrt{x(n+1)}} \cdot\left(1+O\left(\frac{1}{\sqrt{n+1}}\right)\right),$

and summarizing by

$\frac{L_n^{(\alpha)}\left(\frac x n\right)}{n^\alpha}\approx e^\frac x {2n}\cdot\frac{J_\alpha\left(2\sqrt x\right)}{\sqrt x^\alpha},$

where $J_\alpha$ is the Bessel function.

Moreover^{[citation needed]}

$L_n^{(\alpha-n)}(x)\approx e^x\cdot {\alpha\choose n}$

whenever n tends to infinity.

Recurrence relations

The addition formula for Laguerre polynomials:

$L_n^{(\alpha+\beta+1)}(x+y)= \sum_{i=0}^n L_i^{(\alpha)}(x) L_{n-i}^{(\beta)}(y)$.

Laguerre's polynomials satisfy the recurrence relations

$L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!},$

in particular

$L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x)$

and

$L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x),$

or

$L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x);$

moreover

$\begin{align}L_n^{(\alpha)}(x)- \sum_{j=0}^{\Delta-1} {n+\alpha \choose n-j} (-1)^j \frac{x^j}{j!}&= (-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(\alpha+\Delta)}(x)\\[6pt] &=(-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha-i-1 \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x).\end{align}$

They can be used to derive the four 3-point-rules

$\begin{align} L_n^{(\alpha)}(x) & = L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j} L_{n-j}^{(\alpha-k+j)}(x), \\[10pt] n L_n^{(\alpha)}(x) & = (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x), \\[10pt] & \text{or } \frac{x^k}{k!}L_n^{(\alpha)}(x) = \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x), \\[10pt] n L_n^{(\alpha+1)}(x) & =(n-x) L_{n-1}^{(\alpha+1)}(x) + (n+\alpha)L_{n-1}^{(\alpha)}(x) \\[10pt] x L_n^{(\alpha+1)}(x) & = (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x); \end{align}$

combined they give this additional, useful recurrence relations

$\begin{align}L_n^{(\alpha)}(x)&= \left(2+\frac{\alpha-1-x}n \right) L_{n-1}^{(\alpha)}(x)- \left(1+\frac{\alpha-1}n \right) L_{n-2}^{(\alpha)}(x)\\[10pt] &= \frac{\alpha+1-x}n L_{n-1}^{(\alpha+1)}(x)- \frac x n L_{n-2}^{(\alpha+2)}(x). \end{align}$

A somewhat curious identity, valid for integer i and n, is

$\frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x);$

it may be used to derive the partial fraction decomposition

$\frac{L_n^{(\alpha)}(x)}{{n+ \alpha \choose n}}= 1- \sum_{j=1}^n \frac{x^j}{\alpha + j} \frac{L_{n-j}^{(j)}(x)}{(j-1)!}= 1- \sum_{j=1}^n (-1)^j \frac{j}{\alpha + j} {n \choose j}L_n^{(-j)}(x) = 1-x \sum_{i=1}^n \frac{L_{n-i}^{(-\alpha)}(x) L_{i-1}^{(\alpha+1)}(-x)}{\alpha +i}.$

Derivatives of generalized Laguerre polynomials

Differentiating the power series representation of a generalized Laguerre polynomial k times leads to

$\frac{\mathrm d^k}{\mathrm d x^k} L_n^{(\alpha)} (x) = (-1)^k L_{n-k}^{(\alpha+k)} (x)\,;$

moreover, this following equation holds

$\frac{1}{k!} \frac{\mathrm d^k}{\mathrm d x^k} x^\alpha L_n^{(\alpha)} (x) = {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x),$

which generalizes with Cauchy's formula to

$L_n^{(\alpha')}(x) = (\alpha'-\alpha) {\alpha'+ n \choose \alpha'-\alpha} \int_0^x \frac{t^\alpha (x-t)^{\alpha'-\alpha-1}}{x^{\alpha'}} L_n^{(\alpha)}(t)\,dt.$

The derivate with respect to the second variable α has the surprising form

$\frac{\mathrm d}{\mathrm d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}.$

The generalized associated Laguerre polynomials obey the differential equation

$x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0,\,$

which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

$x L_n^{(k) \prime\prime}(x) + (k+1-x)L_n^{(k)\prime}(x) + (n-k) L_n^{(k)}(x)=0,\,$

where $L_n^{(k)}(x)\equiv\frac{d^kL_n(x)}{dx^k}$ for this equation only.

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

In Sturm–Liouville form the differential equation is

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

which shows that Lα
n is an eigenvector for the eigenvalue n.

Orthogonality

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

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

which follows from

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

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

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

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

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

recursively

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

Moreover,

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

in the associated L²[0, ∞)-space.

Turán's inequalities can be derived here, which is

$L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{{\alpha+n-1\choose n-k}}{n{n\choose k}} L_k^{(\alpha-1)}(x)^2>0.$

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

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

Series expansions

Let a function have the (formal) series expansion

$f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).$

Then

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

The series converges in the associated Hilbert space $L^2[0,\infty)$, iff

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

Other examples

Monomials are represented as

$\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x),$

while binomials have the parametrization

${n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).$

This leads directly to

$e^{-\gamma x}= \sum_{i=0}^\infty \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x) \qquad \left(\text{convergent iff }\operatorname{Re}{(\gamma)} > -\frac{1}{2}\right)$

for the exponential function. The incomplete gamma function has the representation

$\Gamma(\alpha,x)=x^\alpha e^{-x} \sum_{i=0}^\infty \frac{L_i^{(\alpha)}(x)}{1+i} \qquad \left(\Re(\alpha)>-1 , x > 0\right).$

Multiplication theorems

Erdélyi gives the following two multiplication theorems

• $t^{n+1+\alpha} e^{(1-t) z} L_n^{(\alpha)}(z t)=\sum_{k=n} {k \choose n}\left(1-\frac 1 t\right)^{k-n} L_k^{(\alpha)}(z),$
• $e^{(1-t)z} L_n^{(\alpha)}(z t)=\sum_{k=0} \frac{(1-t)^k z^k}{k!}L_n^{(\alpha+k)}(z).$

As a contour integral

The polynomials may be expressed in terms of a contour integral

$L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt$

where the contour circles the origin once in a counterclockwise direction.

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials:

$H_{2n}(x) = (-1)^n\ 2^{2n}\ n!\ L_n^{(-1/2)} (x^2)$

and

$H_{2n+1}(x) = (-1)^n\ 2^{2n+1}\ n!\ x\ L_n^{(1/2)} (x^2)$

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.

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as

$L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)$

where $(a)_n$ is the Pochhammer symbol (which in this case represents the rising factorial).

References

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 773, ISBN 978-0486612720, MR 0167642 .

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248 

• B. Spain, M.G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
• Hazewinkel, Michiel, ed. (2001), "Laguerre polynomials", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Eric W. Weisstein, "Laguerre Polynomial", From MathWorld—A Wolfram Web Resource.
• George Arfken and Hans Weber (2000). Mathematical Methods for Physicists. Academic Press. ISBN 0-12-059825-6. 

• S. S. Bayin (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 3.

External links

• Timothy Jones. "The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the Hydrogen Atom". 
• Weisstein, Eric W., "Laguerre polynomial", MathWorld.