Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients ((k; i)) modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0 ≤ i ≤ k. In particular, we prove that this cannot occur if 2 h ≤ k < p - h. This hypothesis can be weakened to 2 h ≤ k < p if we assume, in addition, that the characteristic polynomial of the relation does not have -1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.