14 (Generalised) Laguerre Polynomials
14.1 Definition of Generalised Laguerre Polynomials
The generalised Laguerre Polynomials \(L^{k}_{n}\left(x\right)\) (the Laguerre Polynomials have \(k=0\)) are defined over the Hilbert space \(\mathcal{L}^{2}[0,\infty)\) with weight function \(w\left(x\right)=x^{k}\exp\left(-x\right)\). Hence, the inner product is now defined as
\[\langle f|g\rangle\triangleq\int^{\infty}_{0}f\left(x\right)^{*}g\left(x\right)x^{k}\exp\left(-x\right)\text{d}x. \tag{14.1}\]
- They solve (the generalised) Laguerre’s differential equation
\[xy''+\left(k+1-x\right)y'+ny=0\]
- First few terms:
\[\begin{aligned} L^{k}_{0}\left(x\right) & =1\\ L^{k}_{1}\left(x\right) & =k+1-x\\ L^{k}_{2}\left(x\right) & =\frac{1}{2}\left(x^{2}-2\left(k+2\right)x+\left(k+1\right)\left(k+2\right)\right) \end{aligned}\]
- Rodrigues’ formula:
\[L^{k}_{n}\left(x\right)=\frac{x^{-k}}{n!}\left(\text{d}_{x}-1\right)^{n}x^{n+k}\]
- Recursion formula: given the first two terms, for \(n\ge2\):
\[L^{k}_{n+1}\left(x\right)=\frac{\left(2n+1+k-x\right)L^{k}_{n}\left(x\right)-\left(n+k\right)L^{k}_{n-1}\left(x\right)}{n+1}\]
- Generating function
\[\frac{\exp\left(-tx/\left(1-t\right)\right)}{\left(1-t\right)^{k+1}}=\sum^{\infty}_{n=0}t^{n}L^{k}_{n}\left(x\right)\]
meaning
\[L^{k}_{n}\left(x\right)=\left.\partial^{n}_{t}\frac{\exp\left(-tx/\left(1-t\right)\right)}{\left(1-t\right)^{k+1}}\right|_{t=0}.\]
- Contour integral expression
\[L^{k}_{n}\left(x\right)=\frac{1}{2\pi i}\oint_{C}\frac{\exp\left(-tx/\left(1-t\right)\right)}{\left(1-t\right)^{k+1}t^{n+1}}\text{d}t\]
where \(C\) encircles the origin anticlockwise but omits the essential singularity at 1.
14.2 Orthogonality, Normalisation and Completeness
The Laguerre polynomials are orthonormal with respect to the inner product in Equation 14.1:
\[\int^{\infty}_{0}L^{k}_{n}\left(x\right)L^{k}_{m}\left(x\right)x^{k}\exp\left(-x\right)\text{d}x=\frac{\left(n+k\right)!}{n!}\delta_{nm}.\]
The functions \[\varphi^{k}_{n}\left(x\right)\triangleq x^{k/2}\exp\left(-x/2\right)L^{k}_{n}\left(x\right)\]
form a complete basis for the Hilbert space \(\mathcal{L}^{2}[0,\infty)\).
14.3 QM example: the Hydrogen atom
If you are interested to see where Laguerre polynomials appear in quantum mechanics, please see my notes on that subject [https://www.felixflicker.com/files/PX2132/PX2132_notes_2022.pdf].