What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$?
Define $F$ in the domain where a continuous function exists that behaves so for $x\geqslant z\geqslant 0$. Other behavior of the function can be anything.
However, constructing $F$ it appears there are infinitely many different such functions. They seem to factor through some series of variable transforms applied to a limit object in the system. The set from which the transforms are selected seems to be finite and short, but how can this be proved?
EDIT: B. K-H suggests basically $F(a,x,z,)=\frac{\int_{\,0}^{\,_\infty}\frac{\;y^{x-a}}{e^y}\cdot dy}{\int_{\,0}^{\,_\infty}\frac{\;y^{x-a-z}}{e^y}\cdot dy}$. Variations thereof can be made and work.
Also work various functions based on well known distribution methods, the steplike behavior of limits of differences of complex logarithms and using these to weight output and appropriately shift input of a periodic function where logically necessary.
Are there any other basic methods of constructing functions with the above behavior?
