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:

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

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 \(L_n^\alpha = L_n\):

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

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:

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

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

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

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

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

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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