On the equation over a finite field

Let N be the number of solutions of the equation over the finite field . L. Carlitz found formulas for N when n=3 or 4, m1= =mn=2. In an earlier paper, we obtained formulas for N when , where m=lcm[m1,…,mn], under a certain restriction on the exponents. In this paper, we find formula for N when md>2 and there exists an ℓ such that md(pℓ+1).