# Thread: Recurrence Relations for Laguerre Polynomials

1. ## Recurrence Relations for Laguerre Polynomials

The Laguerre polynomials $\displaystyle L_n(x)$ are given by the explicit formula

$\displaystyle L_n(x)=\sum_{m=0}^{n}(-1)^m\frac{n!}{(n-m)!(m!)^2}x^m$

Recall that the coeﬃcients of the Laguerre polynomials can be calculated using the recurrence relation;

$\displaystyle a_{m+1}=\frac{m-n}{(m+1)^2}a_m$ , $\displaystyle a_0=1$

For ﬁxed n use induction to prove the above summation formula.

I'm not too sure how to approach this question whether its through dummy variables or telescopic sums or whatever.
Help would be much be appreciated. Thank you

2. ## Re: Recurrence Relations for Laguerre Polynomials

Originally Posted by iPod
The Laguerre polynomials $\displaystyle L_n(x)$ are given by the explicit formula

$\displaystyle L_n(x)=\sum_{m=0}^{n}(-1)^m\frac{n!}{(n-m)!(m!)^2}x^m$

Recall that the coeﬃcients of the Laguerre polynomials can be calculated using the recurrence relation;

$\displaystyle a_{m+1}=\frac{m-n}{(m+1)^2}a_m$ , $\displaystyle a_0=1$

For ﬁxed n use induction to prove the above summation formula.

I'm not too sure how to approach this question whether its through dummy variables or telescopic sums or whatever.
Help would be much be appreciated. Thank you
Use induction.

CB