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What you have is basically a nonlinear second order recursion, and in this case it can be solved by: sol = RSolve[-k x f[x] + k (x + 1) f[x + 1] - r f[x] + r f[x - 1] == 0, f[x], x] The answer is fairly large, and besides having variables r and k, it also has two constants C[1] and C[2], so there may be enough flexibility to enforce your desired ...

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