Can you please point out how to build Ackerman function (actually I'm interested in a version proposed by Rózsa Péter and Raphael Robinson) via standard mu-recursive operators? I tried original papers by Péter and Robinson, but Péter's paper use a language different from English and Robinson's papers “Recursion and Double Recursion” and “Primitive Recursive Functions” also don't help: first of them seems more relevant, but is uses so called double recursion operator to define Ackerman function, so in this case explicit definition of the operator in mu-recursive terms is seeked.

Most closely to the answer goes P. Smith in “An introduction to Godel's theorems” (CUP, 2007) (29.4 The Ackermann-Peter function is μ-recursive), but he comes up with the following: “making the argument watertight is pretty tedious though not difficult. There’s nothing to be learnt from spelling out the details here: so we won’t.”

I also tried Rózsa Péter's book “Recursive functions” (1967, Academic press). There are plenty of variants for recursion operators given there. Usually one reduces to other. I believe that there is a type of recursion operator which fit for definition of Ackerman function and sequence of steps which reduce it to primitive redursion and minimisation operators, but I found myself unable to investigate the whole way down.