  We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor
diffusions by a non-Markovian time change in the corresponding Pearson
diffusions. Pearson diffusions are governed by the backward Kolmogorov
equations with space-varying polynomial coefficients and are widely used in
applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor
diffusions are governed by the fractional backward Kolmogorov equations and
have heavy-tailed marginal distributions in the steady state. We derive the
explicit expressions for the transition densities of the fractional reciprocal
gamma and Fisher-Snedecor diffusions and strong solutions of the associated
Cauchy problems for the fractional backward Kolmogorov equation.
