On some permutation polynomials over finite fields

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Let p be prime, q = pm, and q − 1 = 7s. We completely describe the permutation behavior of the binomial P(x) = xr(1 + xes) (1 ≤ e ≤ 6) over a finite field in terms of the sequence {an} defined by the recurrence relation an = an−1 + 2an−2 − an−3 (n ≥ 3) with initial values a0 = 3, a1 = 1, and a2 = 5.