I've read a few times that integration is a sort of extension of summation so my point is: is there any sensated way to extend other operations in the same way ? If this extension exists what branch of math studies it ? It is or could it be useful somewhere ?
as @user17762 answered Multiplicative calculus studies product integrals which are extension of multiplication. Is there other kinds of calculus for the other operations ?
This extension has been studied and is known as product calculus. Look here for product integral.
Yes. In fact, there are an infinite number of Calculus!
Define the generalized mean of a set of number by,
$$(1) \quad \mu_{F}=F^{-1}\left({1 \over N} \cdot \sum_{k=1}^{N} F(x_k)\right)$$
Where $F(x_k)$ is a function over the number. Check that $F(x)=x$ yields the arithmetic mean, $F(x)=\log(x)$ yields the geometric mean, and $F(x)={1 \over x}$ yields the harmonic mean. To define the derivative and integral, we just need a moment of insight to see how manipulating $(1)$ yields standard calculus, given a proper $F$. We find that,
$$(2) \quad {{df} \over {dx}}_F=\lim_{\epsilon \to 0^+} \ F^{-1} \left( {1 \over {\epsilon}} \cdot \left[F(f(x+\epsilon)-F(f(x)) \right] \right)$$ and $$(3) \quad \int_a^b f(x) \ dx_F=\lim_{N \to {\infty}} \ F^{-1}\left({{b-a} \over N} \cdot \sum_{k=1}^{N} F\left(f\left(a+(b-a) \cdot {k \over N}\right)\right)\right)$$
Where $(2)$ is the $F$ derivative and $(3)$ is the $F$ integral. Setting $F(x)=x$ yields standard calculus, $F(x)=\log(x)$ yields product calculus, and $F(x)={1 \over x}$ yields harmonic calculus. At this point speaking of "interpretations" for $(2)$ and $(3)$ given an arbitrary $F(x)$ is naïve. It's best to think in terms of the way the derivative approximates a $f(x)$ around a point. We can manipulate $(2)$ into,
$$(4) \quad f(x+dx)=F^{-1} \left(F(f(x))+F \left({{df} \over {dx}} \right) \cdot dx \right)$$
Once again, regular calculus yields the best linear approximation, product calculus yields the best exponential approximation, and harmonic calculus yields the best rational function approximation.
I should reiterate that this formalism creates an infinite number of calculus, each with their own unique properties and uses.
