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I was wondering if you would point me to a book where the theory of second order homogeneous linear difference equation with variable coefficients is discussed. I am having difficulties in getting rigorous methods to solve some equations, see an example below.

In particular, given the recurrence relation

$X_{n+2} = \frac{3n-2}{n-1}X_{n+1} - \frac{2n}{n-1}X_n$,

two solutions are

$X(n)= n$ and $X(n) = 2^n$.

Is there an "elementary" way of arriving at these solutions? (i.e. without using transforms, etc.)

Thanks in advance.

2 Answers 2

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HINT $\ $ Factor the difference operator. With the shift operator $\rm\ S\ X_n = X_{n+1}\ $ we have

$\rm\ ((n-1)\ S^2 - (3\ n-2)\ S + 2\ n)\ \ X_n\ =\ ((n-1)\ S - n)\ (S - 2)\ \ X_n$ Now put $\rm\ Y_n = (S - 2)\ X_n = X_{n+1} - 2\ X_n\:.\ $ Then the above second-order equation reduces to $\rm\ (n-1)\ Y_{n+1} - n\ Y_n = 0\:.\ $ Solve that for $\rm\:Y_n\:$ and then plug it into the prior equation to obtain a first-order nonhomogeneous equation for $\rm\: X_n\:.$

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    @BillDubuque, could you please provide a reference book for this kind of solutions to this kind of problems?2015-05-21
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I'm not sure this is what you're looking for but do you know of H. Wilf's book "generatingfunctionology"?

He made it available online for free:

http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf

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    Thanks. I was aware of that book. As you know, it does not discuss what I am really after.2011-01-18