Simultaneous Recurrence Relations

Q: I have a set of recurrence relations, say

How can I simplify them into a single recurrence relation involving only , , , ... ?

A: Daniel Lichtblau (danl@wolfram.com) answers: To eliminate , , , and , you need more equations. Write these equations as polynomials, including the polynomial for .

To acquire sufficiently many polynomials to eliminate all and variables, observe that putting gives a total of 7 polynomials in 9 variables.

We want to eliminate 6 variables, elims, from the 7 polynomials.

Here is the desired relation.