  The generalized Fermi-Dirac functions and their derivatives are important in
evaluating the thermodynamic quantities of partially degenerate electrons in
hot dense stellar plasmas. New recursion relations of the generalized
Fermi-Dirac functions have been found. An effective numerical method to
evaluate the derivatives of the generalized Fermi-Dirac functions up to third
order with respect to both degeneracy and temperature is then proposed,
following Aparicio. A Fortran program based on this method, together with a
sample test case, is provided. Accuracy and domain of reliability of some
other, popularly used analytic approximations of the generalized Fermi-Dirac
functions for extreme conditions are investigated and compared with our
results.
