By the same way will have:f(x,y)^[1/p]=p*y/(p+x*y)=g(x,y) p integer
verifying g(x,g(x....)))n times =f(x,y)=y/(1+x*y).
 We may generalize:
     if f(x,y)=phi^-1(phi(y)+n(x))   or  m^[n(x)](y) ,
     phi(m(y))=phi(y)+1 ;
     then f(x,y)^[r]=phi^-1(phi(y)+r.n(x))   or  m^[r.n(x)](y) ,
     here r is a positive real number.

For ideas consider Stephen Wolfram's question:
 > How can one extend recursive function definitions to continuous
 > numbers? What is the continuous analog of the Ackermann function? The
 > symbolic forms of the Ackermann function with a fixed first argument
 > seem to have obvious interpretations for arbitrary real or complex
 > values of the second argument. But is there a general way to extend
 > these kinds of recursive definitions to continuous cases? Given a way
 > to do this, how does it apply to recursive definitions like those on
 > page 130? What happens to all the irregularities when one is between
 > integer values? Or is it only possible to find simple continuous
 > generalizations to functions that show fundamentally simple behavior?
 > Can this be used as a characterization of when the behavior is simple?

For my non-peer-reviewed response to Wolfram's question on the NKS Forum
see http://forum.wolframscience.com/showthread.php?s=&threadid=488

Here's why I'm interested in the subject. A theory of continuously
iterated smooth matrix functions would probably encompass all of
dynamics in physics. A theory of continuously iterated smooth complex
functions would be powerful enough to extend the Ackermann function to
complex numbers.