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Algorithm for obtaining Laguerre Polynomial Coefficients

  1. Jun 26, 2012 #1
    Hi!
    I am trying to prove to myself that the following method gives the nth Laguerre polynomial, defined here as:
    [itex]L_{n}(x)=e^{x}\frac{\partial^{n}}{\partial x^{n}}(x^{n}e^{-x})[/itex]

    I think that the characteristic polynomial of the following n x n matrix will yield the nth Laguerre polynomial.

    [itex]\left[ \begin{array}{cccc} 1 & 1 & 0 & 0 & ... & 0\\ 1 & 3 & 2 & 0 & ... & 0 \\ 0 & 2 & 5 & 3 & ... & 0 \\ 0 & 0 & 3 & 7 & ... & 0 \\ ... & ... & ... & ... & ... & n-1 \\ 0 & 0 & 0 & ... & n-1 & 2n-1\end{array} \right][/itex]

    It is a tridiagonal matrix, with the positive odd integers along the main diagonal up to 2n-1, and the natural numbers up to n-1 in the super and sub diagonals.

    This seems to be an induction proof, but I am having trouble with it. I am wondering if anyone has any ideas on how to proceed. I should be able to recover the recurrence relation for the Laguerre polynomials.

    EDIT: Nevermind! I figured it out! I can use the general formula for the determinant of a tridiagonal matrix! :)
    http://en.wikipedia.org/wiki/Tridiagonal_matrix
     
    Last edited: Jun 27, 2012
  2. jcsd
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