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I'm looking at ways of solving 2nd order difference equations with non-constant coefficients. I am working on a method to use transformations (ie rewriting the equation in new variables) to change the form of the equation. Such as \(\displaystyle a_{n + 2} + f(n) a_{n +1} + g(n) a_n = 0\) to something like \(\displaystyle u_{n + 2} + h(n) u_n = k(n)\) I don't know if this is even possible but I thought I'd give it a try. Anyway, I'm starting with some simple examples and I'm already running into theoretical troubles. Say we have the system:

\(\displaystyle a_{n + 1} = n^2 b_n\) and \(\displaystyle b_{n + 1} = a_n\).

I would like to write something like

\(\displaystyle \left ( \begin{matrix} a_{n + 1} \\ b_{n + 1} \end{matrix} \right ) = \left ( \begin{matrix} 0 & n^2 \\ 1 & 0 \end{matrix} \right ) \left ( \begin{matrix} a_n \\ b_n \end{matrix} \right )\)

I can "solve" this using the usual Linear Algebra methods (diagonalize the matrix, solve the problem in the eigenvector basis, then transform back to the original variables) but the problem is that there is a new connection... I can always make \(\displaystyle a_{n + 1}\) from \(\displaystyle a_n\) just by taking n -> n +1 and that seems to ruin the method. Is there any matrix method I can use to work with this?

In case anyone cares: the solution to \(\displaystyle a_{n + 2} - 3n a_n = 0\) is

\(\displaystyle a_n = - (-1)^{n(n + 1)/2} (-3)^{n/2} (n - 2)! \left ( A + B \sum_{k = 1}^{n - 1} (-1)^k \right )\)

(I'm trying to find a good function to simplify that \(\displaystyle (-1)^{n(n + 1)/2}\). Any suggestions?)

I've been having fun stretching my new skills in Difference Calculus. I already know I've only scratched the surface but I found a set of talks on the internet yesterday on Galois theory of Difference Equations. I only know a teensy touch of Galois theory so I didn't really look at it.

-Dan

\(\displaystyle a_{n + 1} = n^2 b_n\) and \(\displaystyle b_{n + 1} = a_n\).

I would like to write something like

\(\displaystyle \left ( \begin{matrix} a_{n + 1} \\ b_{n + 1} \end{matrix} \right ) = \left ( \begin{matrix} 0 & n^2 \\ 1 & 0 \end{matrix} \right ) \left ( \begin{matrix} a_n \\ b_n \end{matrix} \right )\)

I can "solve" this using the usual Linear Algebra methods (diagonalize the matrix, solve the problem in the eigenvector basis, then transform back to the original variables) but the problem is that there is a new connection... I can always make \(\displaystyle a_{n + 1}\) from \(\displaystyle a_n\) just by taking n -> n +1 and that seems to ruin the method. Is there any matrix method I can use to work with this?

In case anyone cares: the solution to \(\displaystyle a_{n + 2} - 3n a_n = 0\) is

\(\displaystyle a_n = - (-1)^{n(n + 1)/2} (-3)^{n/2} (n - 2)! \left ( A + B \sum_{k = 1}^{n - 1} (-1)^k \right )\)

(I'm trying to find a good function to simplify that \(\displaystyle (-1)^{n(n + 1)/2}\). Any suggestions?)

I've been having fun stretching my new skills in Difference Calculus. I already know I've only scratched the surface but I found a set of talks on the internet yesterday on Galois theory of Difference Equations. I only know a teensy touch of Galois theory so I didn't really look at it.

-Dan

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