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Categories: Mathematics.

Laguerre polynomials

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

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

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

$$\begin{aligned} L_n(x) &= \frac{1}{n!} \exp(x) \dvn{n}{}{x}\big(x^n \exp(-x)\big) \\ &= \frac{1}{n!} \Big( \dv{}{x}- 1 \Big)^n x^n \end{aligned}$$

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

$$\begin{gathered} L_0(x) = 1 \qquad \quad L_1(x) = 1 - x \qquad \quad L_2(x) = \frac{1}{2} (x^2 - 4 x + 2) \end{gathered}$$

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

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

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

$$\begin{aligned} L_n^\alpha(x) &= \frac{1}{n!} x^{-\alpha} \exp(x) \dvn{n}{}{x}\big( x^{n + \alpha} \exp(-x) \big) \\ &= \frac{x^{-\alpha}}{n!} \Big( \dv{}{x}- 1 \Big)^n x^{n + \alpha} \end{aligned}$$

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

$$\begin{aligned} L_0^\alpha(x) = 1 \qquad L_1^\alpha(x) = \alpha + 1 - x \qquad L_2^\alpha(x) = \frac{1}{2} (x^2 - 2 \alpha x - 4 x + \alpha^2 + 3 \alpha + 2) \end{aligned}$$

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

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

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

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

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

$$\begin{aligned} \boxed{ \Inprod{L_m^\alpha}{w L_n^\alpha} = \int_0^\infty L_m^\alpha(x) \: L_n^\alpha(x) \: w(x) \dd{x} = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{nm} } \end{aligned}$$

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

$$\begin{aligned} \boxed{ f(x) = \sum_{n = 0}^\infty a_n L_n^\alpha(x) = \sum_{n = 0}^\infty \frac{\Inprod{L_n}{w f}}{\Inprod{L_n}{w L_n}} L_n^\alpha(x) } \end{aligned}$$

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

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

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