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

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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

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Use induction.

CB