  We extend former results for coherent states on the circle in the literature
in two ways. On the one hand, we show that expectation values of fractional
powers of momentum operators can be computed exactly analytically by means of
Kummer's confluent hypergeometric functions. Earlier, these expectation values
have only been obtained by using suitable estimates. On the other hand, we
consider the Zak transformation not only to map harmonic oscillator coherent
states to coherent states on the circle as it has been discussed before, but we
also use the properties of the Zak transformation to derive a relation between
matrix elements with respect to coherent states in L2(R) and L2(S1). This
provides an alternative way for computing semiclassical matrix elements for
coherent states on the circle. In certain aspects, this method simplifies the
semiclassical computations in particular if one is only interested in the
classical limit, that is the zeroth order term in the semiclassical expansion.
