This density on the circle shares the property with the plain uniform distribution on the circle that the expected value of z^k is ZERO for each integer k with |k|>1. However, when |k|=1 we (unlike plain uniform) do NOT get zero,

--actually... duh... it occurs to me we can also get all well-behaved densities on the circle |z|=1 which have this "remarkable property" by simply making the density be proportional to A + z + 1/z for any constant A>=2. (Proof: fourier series.) So, my particular density which I constructed in a much more complicated way must be expressible as such a formula (that equality is a remarkable identity, I guess).