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Let $q$ be a prime power and $\mathbb{F}_q$ be a finite field with $q$elements. Let $e$ and $d$ be positive integers. In this paper, for $d\geq2$ and$q\equiv1(\mathrm{mod}~ed(d-1))$, we calculate the number of points on analgebraic curve $E_{e,d}:y^e=x^d+ax+b$ over a finite field $\mathbb{F}_q$ interms of $_dF_{d-1}$ Gaussian hypergeometric series with multiplicativecharacters of orders $d$ and $e(d-1)$, and in terms of $_{d-1}F_{d-2}$ Gaussianhypergeometric series with multiplicative characters of orders $ed(d-1)$ and$e(d-1)$. This helps us to express the trace of Frobenius endomorphism of analgebraic curve $E_{e,d}$ over a finite field $\mathbb{F}_q$ in terms of theabove hypergeometric series. As applications, we obtain some transformationsand special values of $_2F_{1}$ Gaussian hypergeometric series.

Hypergeometric functions and algebraic curves $y^e=x^d+ax+b$