Laguerre polynomials are a sequence of polynomials that are solutions to the Laguerre differential equation:
where is a constant.
When , eq420 simplifies to . The solution to this first-order differential equation is , which can be expressed as the Taylor series . This implies that eq420 has a power series solution around . To determine the exact form of the power series solution to eq420, let .
Substituting , and in eq420 yields
Setting in the first sum,
Eq422 is only true if all coefficients of in is 0 (see this article for explanation). So, , or equivalently,
Eq423 is a recurrence relation. If we know the value of , we can use the relation to find .
Recurrence relation | |
Comparing the recurrence relations, we have
where by convention (so that ).
Letting in eq424 and substituting it in yields the Laguerre polynomials:
where we have replaced with .
The first few Laguerre polynomials are: