Laguerre polynomials: Definition from Answers.com

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886), are the canonical 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 Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form $\int_0^\infty f(x) dx$ .

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 to each other with respect to the inner product given by

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

The sequence of Laguerre polynomials is a Sheffer sequence.

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.

1 The first few polynomials
2 Recursive definition
3 Generalized Laguerre polynomials
4 As contour integral
5 Relation to Hermite polynomials
6 Relation to hypergeometric functions
7 Relation to Bessel functions
8 External links
9 Notes
10 References

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:

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

Generalized Laguerre polynomials

The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability density function

$f(x)=\left\{\begin{matrix} e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.$

then

$E \left[ L_n(X)L_m(X) \right]=0\ \mbox{whenever}\ n\neq m.$

The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is, for α > −1,

$f(x)=\left\{\begin{matrix} x^\alpha e^{-x}/\Gamma(1+\alpha) & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.$

(see gamma function) is given by the defining Rodrigues equation for the generalized Laguerre polynomials:

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

These are also sometimes called the associated Laguerre polynomials. 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 using confluent hypergeometric functions and Kummer's transformation as $L_n^{(\alpha)}(x):= {n+ \alpha \choose n} M(-n,\alpha+1,x)={n+ \alpha \choose n} \sum_{i=0} (-1)^i \frac{{n \choose i}}{{\alpha+i \choose i}}x^i\,$ $= e^x\cdot {n+ \alpha \choose n} M(\alpha+n+1,\alpha+1,-x)$ $=\frac{e^x \sin(n \pi )}{\sin((n+\alpha)\pi)}L_{-\alpha-n-1}^{(\alpha)}(-x)$ $=e^x\cdot\sum_{i=0}(-1)^i {\alpha+n+i \choose n} \frac{x^i}{i!}.$ When $n$ is an integer the function reduces to a polynomial of degree $n$ .
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 coefficient of the leading term is (−1)ⁿ/n!;
The constant term, which is the value at the origin, 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 $(0, n+\alpha+ (n-1) \sqrt{n+\alpha}]$ .

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

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

$L_n^{(\alpha)}(-x) \approx \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \exp\left(2 \sqrt{x \left(n+\frac{\alpha+1}{2}\right)} \right)$ . ^[1].

The first few generalized Laguerre polynomials are:

$L_0^{(\alpha)} (x) = 1$

$L_1^{(\alpha)}(x) = -x + \alpha +1$

$L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2}$

$L_3^{(\alpha)}(x) = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2} + \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}$

The explicit formula allows the generalized Laguerre polynomials to be computed using Horner's method:

 function LaguerreL(n, alpha, x) {
    LaguerreL:= 1; bin:= 1 
    for i:= n to 1 step -1 {
        bin:= bin* (alpha+ i)/ (n+ 1- i)
        LaguerreL:= bin- x* LaguerreL/ i
    }
    return LaguerreL;
 }

Recurrence relations

Laguerre's polynomials satisfy the recurrence relations

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

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

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

They can be used to derive the 4 3-point-rules

$L_n^{(\alpha)}(x) = L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x)$ $=\sum_{j=0}^k {k \choose j} L_{n-j}^{(\alpha-k+j)}(x),$
$n L_n^{(\alpha)}(x) = (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x),$ or $\frac{x^k}{k!}L_n^{(\alpha)}(x)= \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x),$
$n L_n^{(\alpha+1)}(x)=(n-x)L_{n-1}^{(\alpha+1)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)$ and
$x L_n^{(\alpha+1)}=(n+\alpha)L_{n-1}^{\alpha}(x)-(n-x)L_n^{(\alpha)}(x);$

combined they give this additional, popular recurrence relation

$L_{n + 1}^{(\alpha)}(x) = \frac{1}{n + 1} \left( (2n + 1 + \alpha - x)L_n^{(\alpha)}(x) - (n + \alpha) L_{n - 1}^{(\alpha)}(x)\right).$

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

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

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 k-th 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 $L_n^{(k)}(x)\equiv\frac{dL_n(x)}{dx^k}$ 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 $L_n^{(k)}(x)=(-1)^k\frac{dL_{n+k}(x)}{dx^k}\,$ , the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.

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

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

The associated, symmetric kernel polynomial has the representations

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

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.

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

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

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

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

A related series expansion is

$f(x)= e^{\frac{\gamma}{1+\gamma} x} \cdot \sum_{i=0} \frac{L_i^{(\alpha)}\left(\frac{x}{1+\gamma}\right)}{(1+\gamma)^{i+\alpha+1}} \sum_{n=0}^i \gamma^{i-n} {i \choose n} f_n^{(\alpha)};$

in particular

$e^{-\gamma x} \cdot L_n^{(\alpha)}(x(1+\gamma))= \sum_{i=n} \frac{L_i^{(\alpha)}(x)}{(1+\gamma)^{i+\alpha+1}} \gamma^{i-n} {i \choose n},$

which follows from

$L_n^{(\alpha)}\left(\frac{x}{1+\gamma} \right)= \frac{1}{(1+\gamma)^n} \sum_{i=0}^n \gamma^{n-i} {n+\alpha \choose n-i} L_i^{(\alpha)}(x).$

Secondly,

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

a consequence derived from

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

for $\operatorname{Re}{(2\alpha- \beta)}>-1$ .

More and other examples

Monomials are representated as

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

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

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

and, even more generally,

$\frac{x^\beta e^{-\gamma x}}{\Gamma(\beta+1)}= {\alpha+\beta \choose \alpha} \sum_{i=0} \frac{L_i^{(\alpha)}(x)}{ {\alpha+i \choose i}} \sum_{j=0}^i \frac{(-1)^j}{(1+\gamma)^{\alpha+ \beta+ j+ 1}} {\alpha+\beta+j \choose j} {\alpha+i \choose i-j}.$

For $β$ a non-negative integer this simplifies to

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

for $γ = 0$ to

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

$\frac{x^\beta L_n^{(\gamma)}(x)}{\Gamma(\beta+1)} = {\alpha+ \beta \choose \alpha} \sum_{i=0} \frac{L_i^{(\alpha)}(x)}{{\alpha+i \choose i}}\sum_{j=0}^n (-1)^{i-j} {n+ \gamma \choose n-j} {\beta+j \choose i} {\alpha+ \beta+ j \choose j}.$

Jacobi's theta function has the representation

$\sum_{k \in \mathbb{Z}} e^{-k^2 \pi x}= \sum_{i=0} L_i^{(\alpha)}\left(\frac{x}{t}\right) \sum_{k \in \mathbb{Z}} \frac{(k^2 \pi t)^i}{(1+ k^2 \pi t)^{i+\alpha+1}};$

the Bessel function $J α$ can be expressed (using an arbitrarily chosen parameter $t$ ) as

$\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{i=0} \frac{L_i^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{{i+ \alpha \choose i}} \frac{t^i}{i!};$

Gamma function has the parametrization

$\Gamma(\alpha)=x^\alpha \sum_{i=0} \frac{L_i^{(\alpha)}(x)}{\alpha+i} \qquad \left(\Re(\alpha)<\frac 1 2\right);$

the lower incomplete Gamma function has the representations

$\frac{\gamma(s;z)}{t^s \Gamma(s)}= \frac{\left(\frac{z}{t}\right)^\alpha}{\Gamma(\alpha+1)} \sum_{i=0} \frac{L_i^{(\alpha)}\left(\frac{z}{t}\right)}{{\alpha+i \choose i}} \sum_{j=0}^i \frac{(-1)^j}{(1+t)^{s+j}}{s-1+j \choose j}{\alpha-1+i \choose i-j},$

$\frac{\gamma(s;z)}{t^s \Gamma(s)}= {\alpha+s \choose \alpha+1} \sum_{i=0} \frac{{\alpha+ i+1\choose i+1}- L_{i+1}^{(\alpha)}\left( \frac{z}{t}\right)}{{\alpha+ i+1\choose i}} \sum_{j=0}^i \frac{(-1)^j}{(1+t)^{\alpha+1+s+j} } {\alpha+s+j \choose j}{\alpha+i+1 \choose i-j}.$

and

$\gamma(s,z)=\frac{\gamma^s}{\Gamma(1-s)} \sum_{i=0} \frac{L_{i+1}^{(-s)}(0)-L_{i+1}^{(-s)}\left(\frac{z}{\gamma}\right)}{(1+\gamma)^{i+1}} \sum_{n=0}^i \gamma^{i-n} \frac{{i \choose n}}{n+1-s};$

The upper incomplete gamma function then is

$\begin{align}\frac{\Gamma(s,z)}{z^s e^{- z}}&= \sum_{k=0} \frac{L_k^{(\alpha)}(z)}{(k+1) {k+1+\alpha-s \choose k+1}} \qquad \left(\Re\left(s-\frac \alpha 2 \right)< \frac 1 4 \right)\\ &= \sum_{k=0} L_k^{(\alpha)}(z\, t) \cdot \frac{_2F_1\left(1+\alpha+k, 1+k; 2+\alpha+k-s; \frac{t-1}{t}\right)}{t^k(k+1){1+\alpha+k-s \choose 1+k}} \\ &= t^s \sum_{k=0} L_k^{(\alpha)}(z\, t) \cdot \frac{_2F_1\left(1-s, 1+\alpha-s; 2+\alpha+k-s; \frac{t-1}{t}\right)}{(k+1){1+\alpha+k-s \choose 1+k}}\\ &= t^{1+\alpha} \sum_{k=0} L_k^{(\alpha)}(z \, t) \cdot \frac{_2F_1\left(1+\alpha+k, 1+\alpha-s; 2+\alpha+k-s; 1-t \right)}{(k+1){1+\alpha+k-s \choose 1+k}},\end{align}$

where $2 F 1$ denotes the hypergeometric function.

As 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).

Relation to Bessel functions

In terms of modified Bessel functions (Bessel polynomials) these following relations hold:

$\begin{align}L_n^{(\alpha)}(x) =& e^\frac x 2 \left(\frac x 4\right)^{n+\frac 12}\frac{2 }{\sqrt \pi (n+1)! {-\frac 1 2 \choose n+1}} \cdot \\ &\cdot\sum_{k=0}^n (-1)^{k+1}{2n+1 \choose n-k} \frac{{n+\alpha \choose n}{\alpha+2n+1 \choose n-k}}{{n-k+\alpha \choose n-k}} \left(k+\frac 1 2 \right) K_{k+\frac 1 2}\left(\frac x 2 \right)\\ = &e^\frac{x}{2} \left(\frac{4}{x} \right)^{n+\alpha+\frac{1}{2}} \Gamma\left(\alpha+\frac{1}{2} \right) {-\alpha-1 \choose n}{-\alpha-\frac 1 2 \choose n} \cdot \\ & \cdot n! \sum_{k=n} \frac{{-2n-1-2\alpha \choose k-n} {-2n-1-\alpha \choose k-n}}{{-\alpha-1 \choose k-n}} \left(\alpha+\frac{1}{2}+k \right) I_{\alpha+\frac 1 2+k} \left(\frac x 2 \right) \end{align},$

or further elaborated

$L_n^{(\alpha)}(x)= \frac 2 {4^n (2n+1) {-\frac 1 2 \choose n}} \sum_{k=0}^n \left(k+\frac 1 2 \right) \frac{{2n+1 \choose n-k}}{{n \choose k}^2} {n+\alpha \choose k}{2n+\alpha+1 \choose n-k} \frac{x^{n-k}}{(n-k)!}L_k^{-2k-1}(x).$

External links

A quick informal derivation of the Laguerre polynomial in the context of the quantum mechanics of hydrogen

Notes

^ Abramowitz, p. 506, 13.3.8

References

Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
B Spain, M G Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
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.

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Laguerre polynomials

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