The Laguerre polynomials are a set of useful functions that arise in physics.
They are the non-singular eigenfunctions of Laguerre’s equation,
with the corresponding eigenvalues being non-negative integers:
The th-order Laguerre polynomial
is given in the form of a Rodrigues’ formula by:
The first couple of Laguerre polynomials are therefore as follows:
Based on Laguerre’s equation,
Laguerre’s generalized equation is as follows,
with an arbitrary real (but usually integer) parameter ,
and still a non-negative integer:
Its solutions, denoted by ,
are the generalized or associated Laguerre polynomials,
which also have a Rodrigues’ formula.
Note that if then :
The first couple of associated Laguerre polynomials are therefore as follows:
And then more can be computed quickly
using the following recurrence relation:
The derivatives are also straightforward to calculate
using the following relation:
Noteworthy is that these polynomials (both normal and associated)
are all mutually orthogonal for ,
with respect to the weight function :
Where is the Kronecker delta.
Moreover, they form a basis in
the Hilbert space
of all functions for which is finite.
Any such can thus be expanded as follows:
Finally, the are related to
the Hermite polynomials like so: