Using the definition of the Hermite polynomials and then integrating by parts $n$ times, we get $f_n(x)=x^n$.

Using the definition of the Hermite polynomials and then integrating by parts $n$ times, we get $f_n(x)=x^n$.

Details: \begin{equation} \begin{aligned} f_n(x)&=\int(-1)^n\eta^{(n)}(x-z)\frac{\eta(z)}{\eta(x-z)}\,dz \\ &=(-1)^ne^{x^2/2}\int\eta^{(n)}(x-z)\ e^{-xz}\,dz \\ &=(-1)^{n-1}e^{x^2/2}x\int\eta^{(n-1)}(x-z)\ e^{-xz}\,dz \\ &\vdots \\ &=e^{x^2/2}x^n\int\eta(x-z)\ e^{-xz}\,dz \\ &=x^n. \end{aligned} \end{equation}