Would there be new insights if hyperops are extended to functions?

[Ivars]

Hi,

Just wondering - sometimes jumping forward and looking back gives new perspective.

If we use f(x) f( b) instead of x, b, we get:

f(x)+f(x)= 2f(x)
f(x)*f(x)= f(x)^2
f(x)^f(x)= f(x)[3]f(x)

f(x)[4]n= ?

h(f(z)^(1/f(z))= f(z)?

d(h(f(z)^(1/f(z))=d(f(z))?

If d(e^z)/dz = e^z, then dh((e^z)^(e-z) ) /dz = d(e^z)/dz =e^z = h(((e^z)^(e-z))?

d(sin z)/dz = cos z, h(sin(z)^(1/sin(z))= sin(z)? dh(sin z ^(1/sin z))/dz = dsin(z)/dz = cos z = h(cosz^(1/cos(z)))?

d(W(z))/dz = W(z)/z(1+W(z))

d(h(W(z)^(1/W(z)))/dz= W(z)/z(1+W(z))?

solutions to f(x) = a^f(x)?

Then if f(x) = const=b we get b[4]n , if f(x)=x we get F(x)= x[4]n , if f(x)=x^2 we have x^2[4]n , if f(x)=e^x we get e^x[4]n

Many generalizations are probably possible. Do they lead anywhere?
I just got the idea while reading about time scale calculus, which generalizes notion of continuos diff equations/difference equations and makes thing unified by replacing difference with graininess which can be a function of point.

Interestingly, in Time scales First order Harmonic numbers (sum(n) 1/n ) is one of the most studied Time scales with certain provable properties.

Excuse me for jumping ahead of events...

Ivars