Generalized Fermi–Dirac functions and derivatives: properties and evaluation

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 [Ap.J.S.S. 117 (1998) 627]. 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.