Categories: Mathematics.

Laguerre polynomials

The Laguerre polynomials are a set of useful functions that arise in physics. They are the non-singular eigenfunctions u(x)u(x) of Laguerre’s equation, with the corresponding eigenvalues nn being non-negative integers:

xu+(1x)u+nu=0\begin{aligned} \boxed{ x u'' + (1 - x) u' + n u = 0 } \end{aligned}

The nnth-order Laguerre polynomial Ln(x)L_n(x) is given in the form of a Rodrigues’ formula by:

Ln(x)=1n!exp(x)dndxn(xnexp(x))=1n!(ddx1)nxn\begin{aligned} L_n(x) &= \frac{1}{n!} \exp(x) \dvn{n}{}{x}\big(x^n \exp(-x)\big) \\ &= \frac{1}{n!} \Big( \dv{}{x}- 1 \Big)^n x^n \end{aligned}

The first couple of Laguerre polynomials Ln(x)L_n(x) are therefore as follows:

L0(x)=1L1(x)=1xL2(x)=12(x24x+2)\begin{gathered} L_0(x) = 1 \qquad \quad L_1(x) = 1 - x \qquad \quad L_2(x) = \frac{1}{2} (x^2 - 4 x + 2) \end{gathered}

Based on Laguerre’s equation, Laguerre’s generalized equation is as follows, with an arbitrary real (but usually integer) parameter α\alpha, and nn still a non-negative integer:

xu+(α+1x)u+nu=0\begin{aligned} \boxed{ x u'' + (\alpha + 1 - x) u' + n u = 0 } \end{aligned}

Its solutions, denoted by Lnα(x)L_n^\alpha(x), are the generalized or associated Laguerre polynomials, which also have a Rodrigues’ formula. Note that if α=0\alpha = 0 then Lnα=LnL_n^\alpha = L_n:

Lnα(x)=1n!xαexp(x)dndxn(xn+αexp(x))=xαn!(ddx1)nxn+α\begin{aligned} L_n^\alpha(x) &= \frac{1}{n!} x^{-\alpha} \exp(x) \dvn{n}{}{x}\big( x^{n + \alpha} \exp(-x) \big) \\ &= \frac{x^{-\alpha}}{n!} \Big( \dv{}{x}- 1 \Big)^n x^{n + \alpha} \end{aligned}

The first couple of associated Laguerre polynomials Lnα(x)L_n^\alpha(x) are therefore as follows:

L0α(x)=1L1α(x)=α+1xL2α(x)=12(x22αx4x+α2+3α+2)\begin{aligned} L_0^\alpha(x) = 1 \qquad L_1^\alpha(x) = \alpha + 1 - x \qquad L_2^\alpha(x) = \frac{1}{2} (x^2 - 2 \alpha x - 4 x + \alpha^2 + 3 \alpha + 2) \end{aligned}

And then more LnαL_n^\alpha can be computed quickly using the following recurrence relation:

Ln+1α(x)=(α+2n+1x)Lnα(x)(α+n)Ln1α(x)n+1\begin{aligned} \boxed{ L_{n + 1}^\alpha(x) = \frac{(\alpha + 2 n + 1 - x) L_n^\alpha(x) - (\alpha + n) L_{n - 1}^\alpha(x)}{n + 1} } \end{aligned}

The derivatives are also straightforward to calculate using the following relation:

dkdxkLnα(x)=(1)kLnkα+k(x)\begin{aligned} \boxed{ \dvn{k}{}{x}L_n^\alpha(x) = (-1)^k L_{n - k}^{\alpha + k}(x) } \end{aligned}

Noteworthy is that these polynomials (both normal and associated) are all mutually orthogonal for x[0,[x \in [0, \infty[, with respect to the weight function w(x)xαexp(x)w(x) \equiv x^\alpha \exp(-x):

Lmα|wLnα=0Lmα(x)Lnα(x)w(x)dx=Γ(n+α+1)n!δnm\begin{aligned} \boxed{ \Inprod{L_m^\alpha}{w L_n^\alpha} = \int_0^\infty L_m^\alpha(x) \: L_n^\alpha(x) \: w(x) \dd{x} = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{nm} } \end{aligned}

Where δnm\delta_{nm} is the Kronecker delta. Moreover, they form a basis in the Hilbert space of all functions f(x)f(x) for which f|wf\Inprod{f}{w f} is finite. Any such ff can thus be expanded as follows:

f(x)=n=0anLnα(x)=n=0Ln|wfLn|wLnLnα(x)\begin{aligned} \boxed{ f(x) = \sum_{n = 0}^\infty a_n L_n^\alpha(x) = \sum_{n = 0}^\infty \frac{\Inprod{L_n}{w f}}{\Inprod{L_n}{w L_n}} L_n^\alpha(x) } \end{aligned}

Finally, the Lnα(x)L_n^\alpha(x) are related to the Hermite polynomials Hn(x)H_n(x) like so:

H2n(x)=(1)n22nn!Ln1/2(x2)H2n+1(x)=(1)n22n+1n!Ln1/2(x2)\begin{aligned} H_{2n(x)} &= (-1)^n 2^{2n} n! \: L_n^{-1/2}(x^2) \\ H_{2n + 1(x)} &= (-1)^n 2^{2n + 1} n! \: L_n^{1/2}(x^2) \end{aligned}