The famous form of an Appell $F_1()$ with two variables looks something like
$F_{1}\left(a,b_1,b_2,c;x_1,x_2\right)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-tx_1)^{-b_1}(1-tx_2)^{-b_2}dt$
I found one confluent hypergeometric function $\Phi()$ that looks something like
$\Phi\left(a,b,c;x_1,x_2\right)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a+N-2}e^{-Ntx_2}(1-t)^{c-a-1}(1-tx_1)^{-b}dt$
Based on the above functions, is there an Apell function $F_{1}()$ of the following form: $F_{1}\left(a,b_1,b_2,c;x_1,x_2\right)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a+N-2}(1-t)^{c-a-1}(1-tx_1)^{-b_1}(1-tx_2)^{-b_2}dt$