  This paper is devoted to the inverse generator problem in the setting of
generators of integrated resolvent operator functions. It is shown that if the
operator $A$ is the generator of a tempered $\beta$-times integrated
$\alpha$-resolvent operator function ($(\alpha,\beta)$-ROF) and it is
injective, then the inverse operator $A^{-1}$ is the generator of a tempered
$(\alpha,\gamma)$-ROF for all $\gamma > \beta+1/2,$ by means of an explicit
representation of the integrated resolvent operator function based in Bessel
functions of first kind. Analytic resolvent operator functions are also
considered, showing that $A^{-1}$ is in addition the generator of a tempered
$(\delta,0)$-ROF for all $\delta<\alpha.$ Moreover, the optimal decay rate of
$(\alpha,\beta)$-ROFs as $t\to \infty$ is given. These result are applied to
fractional Cauchy problem unsolved in the fractional derivative.
