Consider the following set of Fourier transforms (understood as distributions since formally these are divergent): $$ D_{n}(y) := \frac{1}{2\pi} \int_{-\infty}^{\infty} P_{n}(\cos x) e^{i x y} dx $$ where $n \in \mathbb{N}$ and $x>0$, and $P_{n}$ are Legendre Polynomials.
Does there exist a close form expression for $D_{n}$ for general $n$?
One ingredient needed is the representation $$ \delta(y)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ixy} dy $$ of the delta function. With this, since $P_0(\cos x) = 1$ and $P_1(\cos x) = \cos x$ and $P_2(\cos x) = \frac{1}{2}( 3 \cos^2 x - 1 )$, its easily shown that $D_0(y) = \delta(y)$ and $D_{1}(y)= \frac{1}{2} \delta(y - 1) + \frac{1}{2} \delta(y + 1)$ as well as $D_{2}(y) = \frac{1}{4} \delta(y) + \frac{3}{8} \delta(y - 2) +\frac{3}{8} \delta(y + 2)$.
Can one generalize this more simply?
Equivalent to finding the Fourier expansion of $P_n(\cos x)$ (done here by myself): $$P_n(\cos x)=\sum_{k=0}^n b_k b_{n-k}e^{i(n-2k)x},$$ where $b_n=2^{-2n}\binom{2n}{n}=\frac{(2n-1)!!}{(2n)!!}$.