On the rationality of generating functions of certain hypersurfaces over finite fields

Let $a,n$ be positive integers and let $p$ be a prime number. Let ${\mathbb{F}}_q$ be the finite field with $q=p^a$ elements. Let $\{a_i{\}}_{i=1}^{\mathrm{\infty }}$ be an arbitrary given infinite sequence of elements in ${\mathbb{F}}_q$ and $a_1\ne 0$. For each positive integer $i$, let $\{d_{i+j,i}{\}}_{j=0}^{\mathrm{\infty }}$ be an arbitrary given sequence of positive integers with $d_{ii}$ coprime to $q-1$. For each integer $n\ge 1$, let $N_n$, ${\overline{N}}_n$ and ${\tilde{N}}_n$ denote the number of ${\mathbb{F}}_q$-rational points of the hypersurfaces defined by the following three equations:

$$a_1{x}_1+\cdots +a_n{x}_n=b,$$

$$x_1^2+\cdots +x_n^2=b$$

and

$$a_1{x}_1^{d_{11}}+a_2{x}_1^{d_{21}}{x}_2^{d_{22}}+\cdots +a_n{x}_1^{d_{n1}}{x}_2^{d_{n2}}\cdots x_n^{d_{nn}}=b,$$

respectively. In this paper, we show that the generating function $\sum _{n=1}^{\mathrm{\infty }}{N}_n{t}^n$ is a rational function in $t$. Moreover, we show that if $p$ is an odd prime, then the generating functions $\sum _{n=1}^{\mathrm{\infty }}{\overline{N}}_n{t}^n$ and $\sum _{n=1}^{\mathrm{\infty }}{\tilde{N}}_n{t}^n$ are both rational functions in $t$. Moreover, we present the explicit rational expressions of $\sum _{n=1}^{\mathrm{\infty }}{N}_n{t}^n$, $\sum _{n=1}^{\mathrm{\infty }}{\overline{N}}_n{t}^n$ and $\sum _{n=1}^{\mathrm{\infty }}{\tilde{N}}_n{t}^n$, respectively.