I want to calculate this integral with singularity:
$I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}. $
I hope to obtain a closed formula via special functions, maybe some hypergeometric functions. A related differential equation for solving $I(\alpha, x,y)$ as a function of $x$,$y$ is also interesting.