Summary: Let \(D_{\lambda}^{d,k}\) denote the family of diagonal hypersurface over a finite field \(\mathbb{F}_q\) given by \[ D_{\lambda}^{d,k} : X_1^d + X_2^d = \lambda d X_1^k x_2^{d-k}, \] where \(d \geq 2, 1 \leq k \leq d-1\), and \(\gcd (d, k) = 1\). Let \(\# D_{\lambda}^{d,k}\) denote the number of points on \(D_{\lambda}^{d,k}\) in \(\mathbb{P}^1 (\mathbb{F}_q)\). It is easy to see that \(\# D_{\lambda}^{d,k}\) is equal to the number of distinct zeros of the polynomial \(y^d - d \lambda y^k +1\in \mathbb{F}_q [y]\) in \(\mathbb{F}_q\). In this article, we prove that \(\# D_{\lambda}^{d,k}\) is also equal to the number of distinct zeros of the polynomial \(y^{d - k} (1-y)^k - (d\lambda )^{-d}\) in \(\mathbb{F}_q\). We express the number of distinct zeros of the polynomial \(y^{d-k} (1-y )^k - (d\lambda )^{-d}\) in terms of a \(p\)-adic hypergeometric function. Next, we derive summation identities for the \(p\)-adic hypergeometric functions appearing in the expressions for \(\# D_{\lambda}^{d, k}\). Finally, as an application of the summation identities, we prove identities for the trace of Frobenius endomorphism on certain families of elliptic curves.