Let $X \sim \operatorname{Gamma}(\alpha, \beta)$
$\int_{x}^{\infty}(y-x)^{n} d F_{Y}(y)=\int_{x}^{\infty} \sum_{k=0}^{n}\left(\begin{array}{c}{n} \\ {k}\end{array}\right) y^{k}(-x)^{n-k} d F_{Y}(y)$ $=\sum_{k=0}^{n} \int_{x}^{\infty}\left(\begin{array}{l}{n} \\ {k}\end{array}\right) y^{k}(-x)^{n-k} \beta^{\alpha} y^{\alpha-1} \frac{e^{-\beta y}}{\Gamma(\alpha)} d y$ $=\sum_{k=0}^{n} \int_{x}^{\infty}\left(\begin{array}{l}{n} \\ {k}\end{array}\right) y^{k}(-x)^{n-k} \beta^{\alpha}\left(\frac{\beta^{\alpha+k}}{\beta^{\alpha+k}}\right) y^{\alpha-1} \frac{e^{-\beta y}}{\Gamma(\alpha)}\left(\frac{\Gamma(\alpha+k)}{\Gamma(\alpha+k)}\right) d y$ $=\sum_{k=0}^{n}\left(\begin{array}{l}{n} \\ {k}\end{array}\right)(-x)^{n-k} \frac{\beta^{\alpha}}{\beta^{\alpha+k}} \frac{\Gamma(\alpha+k)}{\Gamma(\alpha)} \int_{x}^{\infty} \beta^{\alpha+k} y^{\alpha+k-1} \frac{e^{-\beta y}}{\Gamma(\alpha+k)} d y$ $=\sum_{k=0}^{n}\left(\begin{array}{l}{n} \\ {k}\end{array}\right)(-x)^{n-k} \beta^{-k} \frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}(1-F(x, \alpha+k, \beta)) |$
For odd values of $n-k$ I now get negative values. The result of which seems to be that the values are correct but the sign is flipped. This seems to happen in particular for odd values of $n$. However $E[Y - X | Y > X] > 0$ by definition. What am I missing here?