I am trying to understand this document (https://www.math.uvic.ca/faculty/jing/222generating.pdf), which describes how to derive a generating function from a recurrence relation, and then how to use the generating function to derive a formula for each term in the original recurrence relation.
Near the top of the second page we see the following:
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I feel pretty sure that $A(1 + \beta x) + B(1 + \alpha x)$ should actually be $A(1 - \beta x) + B(1 - \alpha x)$. This is just simple algebra. Am I right about this?
(I see that it doesn't affect the computations because $-\beta A - \alpha B = 0$ is equivalent to $\beta A + \alpha B = 0$, which is what the document says at the bottom.)
User saulspatz says in a comment that I am correct. I can't see any reason why I'd be wrong about this, since it just has to do with giving two fractions a common denominator, so I'm writing this answer just so this question has an accepted answer.
(Note that I have to wait two days before I can accept my own answer.)