Generalized phi(s,a,b,c)

[tommy1729]

To clarify 

let 
f(s+1) = exp(f(s)) + a*s + b
g(s+1) = exp(g(s)) - a*s - b

then 

f(s+1) + g(s+1) = exp(f(s)) + exp(g(s))

Now assume g(s+1) = g(s) ( g is then no longer entire but may be analytic )

For some a and b and f and g this might be interesting.
Or use infinitesimals.

I know not very formal, just sketchy ideas.

Another crazy idea is the generalization with similar functions a,b,c :

a(s+1) + b(s+1) + c(s+1) = exp(a(s+1)) + exp(b(s+1)) + exp(c(s+1))
a(s+2) + b(s+2) + c(s+2) = exp^[2](a(s+1)) + exp^[2](b(s+1)) + exp^[2](c(s+1))

D(s) = a(s) + b(s) + c(s).

And then somehow get tetration from D(s).

Im talking analytic tetration here ofcourse.

crazy ideas :p

regards

tommy1729