The $L^{(\alpha)}$-harmonic function is the solution
of the parabolic operator
$L^{(\alpha)}= \partial_{t}+(-\Delta_{x})^{\alpha}$.
We study a function space $\widetilde{{\cal B}}_{\alpha}(\sigma)$
consisting of $L^{(\alpha)}$-harmonic functions of parabolic Bloch type.
In particular, we give a reproducing formula for functions
in $\widetilde{{\cal B}}_{\alpha}(\sigma)$.
Furthermore, we study the fractional calculus on
$\widetilde{{\cal B}}_{\alpha}(\sigma)$.
As an application, we also give a reproducing formula
with fractional orders for functions
in $\widetilde{{\cal B}}_{\alpha}(\sigma)$.
Moreover, we investigate the dual and pre-dual spaces of
function spaces of parabolic Bloch type.