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Math Help - Lagurre Polynomials

  1. #1
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    Lagurre Polynomials

    Hello all,

    Q_{\ell} ( \ell \in \mathbb{N}_{0}) are the Lagurre polynomials. I would like to derive the following expression:

    Q_{\ell}(x)=\ell! \sum_{k=0}^{\ell}\frac{(-1)^{k}}{(k!)^{2}(l-k)!}x^{k}

    So far I have shown that the coefficients in any power series solution u(x)=\sum_{k=0}^{\infty}c_{k}x^{k} of x\frac{d^{2}u}{du^{2}}+(1-x)\frac{du}{dx}+\ell u=0 will satisfy that:

    c_{k+1}=\frac{k-l}{(k+1)^{2}}c_{k}

    Furthermore I have shown that c_{\ell+1}=c_{\ell+2}=\ldots=0

    Now, using indcution, I would like to show that:

    c_{k}=\frac{\ell!(-1)^{k}}{(k!)^{2}(\ell-k)!} \hspace{1cm}(*)

    I have that c_{0}=1 for k=0. Thus the basis for the induction is established. Now I assume that (*) is true for a fixed k and want to derive the case for k+1. I have done the following:

    c_{k+1}=\frac{\ell!(-1)^{k+1}}{((k+1)!)^{2}(\ell-(k+1))!} =\frac{\ell!(-1)^{k}(-1)^{1}}{(k+1)^{2}(k!)^{2}((\ell-k)-1)!}

    I am stuck here. Suggestions would be greatly appreciated.

    Thanks.
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  2. #2
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    Quote Originally Posted by surjective View Post
    using induction, I would like to show that:

    c_{k}=\frac{\ell!(-1)^{k}}{(k!)^{2}(\ell-k)!} \hspace{1cm}(*)

    I have that c_{0}=1 for k=0. Thus the basis for the induction is established. Now I assume that (*) is true for a fixed k and want to derive the case for k+1. I have done the following:

    c_{k+1}=\frac{\ell!(-1)^{k+1}}{((k+1)!)^{2}(\ell-(k+1))!} =\frac{\ell!(-1)^{k}(-1)^{1}}{(k+1)^{2}(k!)^{2}((\ell-k)-1)!}

    I am stuck here. Suggestions would be greatly appreciated.
    You're almost there! Notice that ((l-k)-1)! = \frac{(l-k)!}{l-k}. So your expression for c_{k+1} is equal to \frac{\l!(-1)^{k}(-1)(l-k)}{(k+1)^{2}(k!)^{2}(l-k)!} = \frac{k-l}{(k+1)^2}c_k , as required.

    Note 1. Instead of starting with the desired formula for c_{k+1} and then showing that it is equal to \frac{k-l}{(k-1)^2}c_k, it is more usual when writing out a proof by induction to reverse that process. In other words, start with the formula (*) for c_k and then show that \frac{k-l}{(k-1)^2}c_k is equal to the desired formula for c_{k+1}.

    Note 2. Laguerre's name has an e after the u.
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