  Anti-circulant tensors have applications in exponential data fitting. They
are special Hankel tensors. In this paper, we extend the definition of
anti-circulant tensors to generalized anti-circulant tensors by introducing a
circulant index $r$ such that the entries of the generating vector of a Hankel
tensor are circulant with module $r$. In the special case when $r =n$, where
$n$ is the dimension of the Hankel tensor, the generalized anticirculant tensor
reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors
are still special Hankel tensors. For the cases that $GCD(m, r) =1$, $GCD(m, r)
= 2$ and some other cases, including the matrix case that $m=2$, we give
necessary and sufficient conditions for positive semi-definiteness of even
order generalized anti-circulant tensors, and show that in these cases, they
are SOS tensors. This shows that, in these cases, there are no PNS (positive
semidefinite tensors which are not sum of squares) Hankel tensors.
