# Recurrence Relations for Laguerre Polynomials

• Nov 24th 2011, 12:08 PM
iPod
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
• Nov 24th 2011, 09:41 PM
CaptainBlack
Re: Recurrence Relations for Laguerre Polynomials
Quote:

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