Fermi-Dirac and Bose-Einstein functions of negative integer order

We find simple explicit closed-form formulas for the Fermi-Dirac function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F}_{ - n} (z) $$\end{document} and Bose-Einstein function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}_{ - n} (z) $$\end{document} for arbitrary n ε 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.