Fourier Transform of Distribution Equal to the Distribution Itself

Question

We define $T(t)=a\delta^{(n)}+bt^n$ for $a, b$ nonzero complex constants and $n$ a nonnegative integer. I want to find the combinations of $a, b, n$ such that the fourier transform of $T$ is equal to $T$ itself. In my mind, the natural first idea is to manually compute the transform. However, I don't know how to find the transform of $t^n$. Is there a way to compare the distribution $T$ to its transform without actually computing the transform? Any help would be appreciated.