The number of solutions of diagonal cubic equations over Galois rings GR(p2,r)

Let $R=GR(p^2,r)$ be a Galois ring of characteristic $p^2$ with cardinality $p^{2r}$, where p is a prime. Let $A_n\left(z\right)$, $B_n\left(z\right)$, $A_n^{\prime }\left(z\right)$ and $B_n^{\prime }\left(z\right)$ denote the number of solutions of the equations $x_1^3+x_2^3+\dots +x_n^3=z$, $x_1^3+x_2^3+\dots +x_n^3+z{x}_{n+1}^3=0$, $p{x}_1^3+p{x}_2^3+\dots +p{x}_n^3=z$ and $p{x}_1^3+p{x}_2^3+\dots +p{x}_n^3+z{x}_{n+1}^3=0$, respectively. In this paper, we show that for any $z\in R$, the generating functions ${\sum }_{n=1}^{\infty }{A}_n\left(z\right)x^n$, ${\sum }_{n=1}^{\infty }{B}_n\left(z\right)x^n$, ${\sum }_{n=1}^{\infty }{A}_n^{\prime }\left(z\right)x^n$ and ${\sum }_{n=1}^{\infty }{B}_n^{\prime }\left(z\right)x^n$ are rational functions in x, and also give explicit expressions for them.