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}^\infty $ be an arbitrary given infinite sequence of elements in $ \mathbb F_q $ and $ a_1\neq 0 $. For each positive integer $ i $, let $ \{d_{i+j, i}\}_{j = 0}^\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 $, $ \bar N_n $ and $ \widetilde{N}_n $ denote the number of $ \mathbb F_q $-rational points of the hypersurfaces defined by the following three equations:

$ a_1x_1+\cdots+a_nx_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}^{\infty}N_nt^n $ is a rational function in $ t $. Moreover, we show that if $ p $ is an odd prime, then the generating functions $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $ are both rational functions in $ t $. Moreover, we present the explicit rational expressions of $ \sum_{n = 1}^{\infty}N_nt^n $, $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $, respectively.