The notion of generating functions and its application to solving recursive equations are very well-known. For reader who did not have a chance to get familiar with the topics, I recommend to take a look at very good book: Concrete Mathematics: A Foundation for Computer Science, by Ronald L. Graham, Donald E. Knuth, Oren Patashnik.
Generating functions are usually applied to single variable recursive equations. But actually, the technique may be extended to multivariate recursive equations, or even to a system of recursive equations.

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