In this paper, we introduce a new subclass of analytic and bi-univalent functions in the open unit disc $U.$ For this subclass of functions, estimates of the initial coefficients $\left|A_2\right|$ and $\left|A_3\right|$ of the Taylor-Maclaurin series are given. An application of Legendre polynomials to this subclass of functions is presented. Furthermore, our study discusses several special cases.