Laguerre polynomials

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

xy'' + (1 - x)y' + ny = 0

which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.

More generally, the name Laguerre polynomials is used for solutions of

xy'' + (\alpha+1 - x)y' + ny = 0~.

Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonin polynomials, after their inventor^[1] Nikolay Yakovlevich Sonin).

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)={\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

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

The sequence of Laguerre polynomials $n! L n$ is a Sheffer sequence,

\frac{d}{dx} L_n = \left ( \frac{d}{dx} - 1 \right ) L_{n-1}.

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 which 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.)

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, closed form, and generating function

One 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{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}.

The closed form is

L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .

The generating function for them likewise follows,

\sum_n^\infty t^n L_n(x)= \frac{1}{1-t} e^{-\frac{tx}{1-t}}.

Polynomials of negative index can be expressed using the ones with positive index:

L_{{-n}}(x)=e^{x}L_{{n-1}}(-x).

Generalized Laguerre polynomials

For arbitrary real α the polynomial solutions of the differential equation ^[2]

x\,y'' + (\alpha +1 - x)\,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

L^{(\alpha)}_0(x) = 1

L^{(\alpha)}_1(x) = 1 + \alpha - x

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

L^{(\alpha)}_{k + 1}(x) = \frac{(2k + 1 + \alpha - x)L^{(\alpha)}_k(x) - (k + \alpha) L^{(\alpha)}_{k - 1}(x)}{k + 1}.

The simple Laguerre polynomials are the special case $α = 0$ of the generalized Laguerre polynomials:

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

The Rodrigues formula for them is

\begin{align} L_n^{(\alpha)}(x) &= {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) \\ &= x^{-\alpha} \frac{( \frac{d}{dx}-1) ^n}{n!}x^{n+\alpha}. \end{align}

The generating function for them is

\sum_n^\infty t^n L^{(\alpha)}_n(x)= \frac{1}{(1-t)^{\alpha+1}} e^{-\frac{tx}{1-t}}.

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

Explicit examples and properties of the generalized Laguerre polynomials

Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as^[3]

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

{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]

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]

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.

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)};

If α is non-negative, then 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 $\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \right].$
The polynomials' asymptotic behaviour for large $n$ , but fixed $α$ and $x > 0$ , is given by^[6]^[7]

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.

As a contour integral

Given the generating function specified above, the polynomials may be expressed in terms of a contour integral

L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint_C\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.

Recurrence relations

The addition formula for Laguerre polynomials:^[8]

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),

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}

Since $L_{n}^{{(\alpha )}}(x)$ is a monic polynomial of degree $n$ in $\alpha$ , there is the partial fraction decomposition

{\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 $L_{n}^{{(\alpha )}}(x)$ in terms of Charlier polynomials:

{\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 of generalized Laguerre polynomials

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

{\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

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.

Moreover, the following equation holds:

\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),

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 derivative with respect to the second variable $α$ has the form,^[9]

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

This is evident from the contour integral representation below.

The generalized 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,

xL_{n}^{[k]\prime \prime }(x)+(k+1-x)L_{n}^{[k]\prime }(x)+(n-k)L_{n}^{[k]}(x)=0,

where $L_{n}^{[k]}(x)\equiv {\frac {d^{k}L_{n}(x)}{dx^{k}}}$ for this equation only.

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 $L (α) n$ is an eigenvector for the eigenvalue $n$ .

Orthogonality

The generalized Laguerre polynomials are orthogonal over $[0, \infty)$ with respect to the measure with weighting function $x α e - x$ :^[10]

\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 $\Gamma(x,\alpha+1,1)$ denotes 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)

\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)\to \delta (y-\cdot ).

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)} (x)\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 $L 2 [0, \infty)$ if and only if

\| 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

\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 \text{convergent iff } \Re(\gamma) > -\tfrac{1}{2}

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 ^[11]

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).

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials:

\begin{align} H_{2n}(x) &= (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2) \\ H_{2n+1}(x) &= (-1)^n 2^{2n+1} n! x L_n^{(1/2)} (x^2) \end{align}

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).

Poisson kernel

\sum_{n=0}^{\infty}\frac{n!L_{n}^{(\alpha)}(x)L_{n}^{(\alpha)}(y)r^{n}}{\Gamma\left(1+\alpha+n\right)}=\frac{\exp\left(-\frac{\left(x+y\right)r}{1-r}\right)I_{\alpha}\left(\frac{2\sqrt{xyr}}{1-r}\right)}{\left(xyr\right)^{\frac{\alpha}{2}}\left(1-r\right)},\qquad \alpha>-1,\quad |r|<1.

Notes

↑ Nikolay Sonin (1880). "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries". Math. Ann. 16 (1): 1–80. doi:10.1007/BF01459227.
↑ A&S p. 781
↑ A&S p.509
↑ A&S p.510
↑ A&S p. 775
↑ G. Szegő, "Orthogonal polynomials", 4th edition, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975, p. 198.
↑ 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
↑ A&S equation (22.12.6), p. 785
↑ W. Koepf, "Identities for families of orthogonal polynomials and special functions.", Integral Transforms and Special Functions 5, (1997) pp.69-102. (Theorem 10)
↑ http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
↑ C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions", Proceedings of the National Academy of Sciences, Mathematics, (1950) pp.752-757.

References

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.
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.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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

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