Laguerre polynomials )
are a sequence of polynomials that are solutions to the Laguerre differential equation:
\frac{dv}{dx}+nv=0\;\;\;\;\;\;\;\;420)
where 
is a constant.
When 
, eq420 simplifies to 
. The solution to this first-order differential equation is =Ae^{-nx})
, which can be expressed as the Taylor series =A\sum_{n=0}^{\infty}\frac{(-nx)^n}{n!})
. This implies that eq420 has a power series solution around . To determine the exact form of the power series solution to eq420, let =\sum_{j=0}^{\infty}a_jx^j)
.
Substituting 
, 
and 
in eq420 yields
a_jx^{j-2}+(1-x)\sum_{j=0}^{\infty}ja_jx^{j-1}+n\sum_{j=0}^{\infty}a_jx^{j}=0\;\;\;\;\;\;\;\;421)
a_jx^{j}=0)
Setting 
in the first sum,
^2a_{j+1}x^{j}+\sum_{j=0}^{\infty}(n-j)a_jx^{j}=\sum_{j=0}^{\infty}(j+1)^2a_{j+1}x^{j}+\sum_{j=0}^{\infty}(n-j)a_jx^{j}=0)
^2a_{j+1}+(n-j)a_j\biggr\]x^{j}=0\;\;\;\;\;\;\;\;422)
Eq422 is only true if all coefficients of 
in is 0 (see this article for explanation). So, ^2a_{j+1}+(n-j)a_j=0)
, or equivalently,
^2}a_j\;\;\;\;\;\;\;\;423)
Eq423 is a recurrence relation. If we know the value of 
, we can use the relation to find 
.
Comparing the recurrence relations, we have
^k\frac{n(n-1)(n-2)\cdots (n-k+1)}{1^22^2\cdots k^2}a_0)
^k\frac{n!}{(k!)^2(n-k)!}\;\;\;\;\;\;\;\;424)
where 
by convention (so that =1)
).
Letting 
in eq424 and substituting it in =\sum_{j=0}^{n}a_jx^j)
yields the Laguerre polynomials:
=\sum_{j=0}^{n}(-1)^j\frac{n!}{(j!)^2(n-j)!}x^j\;\;\;\;\;\;\;\;425)
where we have replaced with )
.
The first few Laguerre polynomials are:
=1)
=-x+1)
=\frac{1}{2}(x^2-4x+2))
=\frac{1}{6}(-x^3+9x^2-18x+6))