Therefore, solutions shall have the form .
The equation associated with finding is: .
First we do the case when .
We looking for a solution .
Substituting that into the equation we get,
We can rewrite this as,
Evaluate the middle summation at and combine,
This tells us that .
The condition in the middle says,
As a consequence
Taking to be arbitrary (like ) shall produce coefficients for .
That would be one solution to the differencial equation.
Of course the other linearly independent solution is found with .