On the equation x1m1+⋯+xnmn=ax1⋯xn over a finite field

Let N be the number of solutions of the equationx1m1+⋯+xnmn=ax1⋯xn over the finite field Fq=FpsFq=Fps. L. Carlitz found formulas for N   when n=3n=3 or 4, m1=⋯=mn=2m1=⋯=mn=2. In an earlier paper, we obtained formulas for N   when d=gcd(∑j=1nm/mj−m,(q−1)/m)=1, where m=lcm[m1,…,mn]m=lcm[m1,…,mn], under a certain restriction on the exponents. In this paper, we find formula for N   when md>2md>2 and there exists an ℓ   such that md|(pℓ+1)md|(pℓ+1).