Two new series for the Fermi–Dirac integral

We derive two new series expressions for the Fermi–Dirac integral Fj(x)Fj(x) based on the Mittag–Leffler expansion of sech(z)sech(z). The first series is exact for all values of x   with j>(−1)j>(−1). The second series is exact for all values of x   and for half-integral values of j⩾(1/2)j⩾(1/2). These expressions will be of use while setting up approximate short expressions for the integral. They also will be useful to interpret the results of Fermi–Dirac distribution in physical phenomena.