,Fermi–Dirac and Bose–Einstein functions of negative integer order

We find simple explicit closed-form formulas for the Fermi–Dirac function $\mathscr{F}_{-n}(z)$ and Bose–Einstein function $\mathscr{B}_{-n}(z)$ for arbitrary $n\in\mathbb{N}$. The obtained formulas involve the higher tangent numbers defined by Carlitz and Scoville. We present some examples and direct consequences of applying the main results.