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.
Best Answer
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 $\rm\: X_n\:.$