The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases of the generalized hypergeometric function $_qF_p$. The Appell hypergeometric functions $F_q$, $q=1,2,3,4$ are product of two hypergeometric functions $_2F_1$ that appear in many areas of mathematical physics. Here, we are interested in the Appell hypergeometric function $F_2$ which is known to have a double integral representation. As demonstrated by Opps, Saad, and Srivastava (J. Math. Anal. Appl. 302 (2005) 180-195), the double integral representation of $F_2$ can be reduced to a single integral that can be easily evaluated for certain values of the parameters in terms of $_2F_1$ and $_3F_2$. Using many of the reduction formulas of $_2F_1$ and $_3F_2$ and the representation of $F_2$ in terms of a single integral, we have begun to tabulate new reduction formulas for $F_2$.