SPECIAL FUNCTIONS
L 0
L 1
L 2
L 3
123 4
5
567
− 5
10
− 10
xFigure 18.7 The first four Laguerre polynomials.18.7.1 Properties of Laguerre polynomialsThe Laguerre polynomials and functions derived from them are important in
the analysis of the quantum mechanical behaviour of some physical systems. We
therefore briefly outline their useful properties in this section.
Rodrigues’ formulaThe Laguerre polynomials can be expressed in terms of a Rodrigues’ formula
given by
Ln(x)=ex
n!dn
dxn(
xne−x)
, (18.112)which may be proved straightforwardly by calculating thenth derivative explicitly
using Leibnitz’ theorem and comparing the result with (18.111). This is illustrated
in the following example.