Can you please point out how to build Ackerman function (actually I'm interested in a version proposed by Rzsa Pter and Raphael Robinson) via standard mu-recursive operators? I tried original papers by Pter and Robinson, but Pter'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 -recursive), but he comes up with the following: making the argument watertight is pretty tedious though not difficult. Theres nothing to be learnt from spelling out the details here: so we wont.I also tried Rzsa Pter'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.
Explicit mu-recursive expression for Ackerman function
computability
Breaking the Ackermann function all the way down to the elementary operators would really be quite lengthy, but here is a sketch:Note that when computing $A(m,x)$ recursively, at any point of the computation you are dealing with an expression of the form $A(m_1,A(m_2,\dots,A(m_k,z)\dots)$. Given a bijective pairing function $p$ with inverse $(\pi_1,\pi_2)$, we can encode this state as $p(z,p(k,p(m_k,\dots,p(m_2,m_1)\dots)$ (just $p(z,0)$ in case $k=0$). We then can define the one-step evaluation function, given a state:$e(p(z,0)) = p(z,0)$;$e(p(z,p(k,p(0,c)))) = p(z+1,p(k-1,c))$;$e(p(0,p(k,p(m+1,c)))) = p(1,p(k,p(m,c)))$;$e(p(z+1,p(k,p(m+1,c)))) = p(z,p(k+1,p(m+1,p(m,c))))$.You then get the n-step evaluation function using primitive recursion:$E(0,m,x) = p(x,p(1,m))$ and $E(n+1,m,x) = e(E(n,m,x))$.Finally, wrap $\mu$-recursion around $E$ to find the point where we get to a state of the form $p(z,0)$  - $z$ will be $A(m,x)$.