ack.tex

\documentclass{article}
\usepackage{expl3}

\ExplSyntaxOn
% \my_ack:nn { <m> } { <n> }
\cs_new:Npn \my_ack:nn #1#2
  {
    \int_compare:nNnTF { #1 } = { \c_zero_int }
      { % m = 0
        \int_eval:n { #2 + 1 }
      }
      {
        \int_compare:nNnTF { #2 } = { \c_zero_int }
          { % n = 0
            \my_ack:ff { #1 - 1 } { \c_one_int }
          }
          { % otherwise
            \my_ack:ff { #1 - 1 } { \my_ack:ff { #1 } { #2 - 1 } }
          }
      }
  }
\cs_generate_variant:Nn \my_ack:nn { ff }
\cs_new_eq:NN \Ack \my_ack:ff
\ExplSyntaxOff

\begin{document}

\[
\mathrm{Ack}(1, 2) = \Ack{1}{2}
\]

\end{document}
% vim: ft=expl3: