The signs of three-term recurrence sequences

In this note, we present two sufficient conditions for determining the signs of three-term recurrence sequences. In order to determine the signs of some sequences, by our method, it suffices to compute a constant number of terms at the beginning. For example, in order to prove the positivity of the central Delannoy number D(n)D(n), by our method, it just needs to know the recurrence relation of D(n)D(n) and the values of D(k)D(k) for 0≤k≤20≤k≤2. As applications, we determine the signs of some famous sequences.

► We present two sufficient conditions for determining the signs of three-term recurrence sequences.
► By our method, in order to determine the signs of some sequences, it suffices to compute a constant number of terms at the beginning.
► We also give some examples to explain our method.