In a Discrete Mathematics video, recurrence relations are solved by applying generating functions to each term, doing algebra, and extracting coefficients of the result. There is no mention of Z-Transforms.
In a Discrete-Time Signals and Systems course, they are solved by applying Z-Transforms to each term, doing algebra in the Z-domain, and undoing the Z-Transforms. This feels very similar to the generating function approach, but there is almost no overlap in the terminology used. The biggest difference appears to be that the Z-Transform uses a negative exponent while the Discrete Mathematics approach uses a positive exponent in the summation.
To what extent is the theory behind these approaches interchangeable? Are they mainly different languages to describe the same thing, or different pieces of mathematics with surface level points of contact?