Nearly linear recurrences generalise linear recurrences and can be represented as special cases of both linear time-invariant systems in control theory and linear-constraint loops in program analysis. We formulate the Positivity Problem for such recurrences: given a recurrence and initial values, decide whether every sequence satisfying the recurrence is termwise nonnegative. This problem generalises the Positivity Problem for linear recurrence sequences and is a special case of halfspace non-reachability for linear time-invariant systems. Our main result is a decision procedure for order-2 recurrences. The termination of our procedure relies on a transcendence theorem of independent interest: we prove that certain convergent series obtained by summing the absolute values of terms of algebraic linear recurrence sequences are transcendental.

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