Solving f( h(x) , y , z- x )=f(x, y, z) ; f(x, y, z +1)= m(f(x ,y, z)

From: Alain Verghote (alainverghote_at_yahoo.fr)
Date: 11/15/04 

Date: 15 Nov 2004 15:51:33 -0500

Dear Friends,

This kind of thing is in the air,I want to show a solving method for
a multidimensionnal functional equation sit on a threefeet stool:
     -Abel counting phi function for h(x)
       phi(h(x))=phi(x)+1
     -power iterated functons like
       m(L)^[r] r real or complex,
     -evolution process
       when (x ,y ,z)-> (h(x) ,y ,z-x) f->f 1)
            (x ,y, z)-> (x ,y,z+1) f-> m(f ) 2)
  So we observe that (y*phi(x)+z+c ) is invariant for
  transformation 1) c any constant , A)
  and transformation 2) taken into account by m^[..z] B)
  combining A) and B) we've got:
          f(x ,y ,z)= m^[y*phi(x)+z+c](L)
 
 Alain.