What is Summability Calculus? I came across a book called "Summability Calculus, The first book in the literature, which is devoted to fractional finite sums as an object of study on its own right." ( See: https://www.springer.com/gp/book/9783319746470 )
What is Summability Calculus? Is it an entirely new field? Or is it a part of Discrete Mathematics / Discrete Calculus? Fields which study finite sums. What justifies the new name? 
 A: Well, it is pretty much what, or at least the cited paper author's approach to what, I have referred to in prior explorations as continuum sums.
Basically, the question involved is this: "If you have a sequence $a_n$ which is defined by a function $f(n)$ on the real numbers, i.e.
$$a_n := f(n)$$
then is there some way to find a real function $F(n)$ such that for the sequence of sums
$$S_n := \sum_{k=0}^{n-1} a_k$$
we have $S_n = F(n)$ when $n$ is a natural number?"
Or, in other words, "how can you interpolate the discrete summation to a continuous function of a real parameter, i.e. assign a meaningful value to
$$\sum_{k=0}^{n-1} f(k)$$
when the summation bound $n$ is not a natural number, but instead a non-natural real number, like $\pi$? E.g. what is $\sum_{k=0}^{\pi} k$?"
As with any other interpolation problem, you need to put constraints on it, or there will be an (uncountable!) infinity of possible answers. The paper, then, is basically discussing a set of constraints that permit the sum to attain, for a reasonably wide class of functions $f$, a well-defined value, and moreover such that this value agrees with, and thus generalizes, many other special cases where that such an interpolation is "naturally" possible, such as how the identity
$$\sum_{k=0}^{n-1} k = \frac{n(n-1)}{2}$$
perhaps familiar from introductory Calculus, can be used to define
$$\sum_{k=0}^{\pi} k := \frac{\pi(\pi+1)}{2} \approx 6.50559.$$
even though directly having a "fractional number of terms" in the sum did not make sense to begin with.
A: Just to add to what @The_Sympathizer mentioned:
First, the case where we have $\sum_{k=0}^{n-1} f(k)$ is the simplest case. What is interesting is to consider the more general case where we have sums of the form $\sum_{k=0}^{n-1} f(k,n)$. The book refers to those as composite sums. For example, what is $\sum_{k=0}^{n-1} \cos(k/n)$ for real values of $n$?
Second, there is a deep connection between fractional finite sums and summability theory of divergent series. It turned out that the definition provided by summability theory coincides with the other definitions, so they can be used interchangeably. 
Third, the summand $f(k)$ does not have to be real-valued. The author looks into complex-valued functions as well in later chapters. 
Fourth, the last chapter looks into the case where the summand $f(k)$ is not differentiable. Even in that case, there is a natural definition as well for fractional finite sums using finite differences. This (again) coincides with the other definitions when the summand happens to be differentiable. 
Fifth, the main argument of the book is that you can deal with fractional finite sums without having to define them in explicit form. For example, you can differentiate, compute Taylor series, derive asymptotic expansions, evaluate, and so on, all using $\sum_{k=0}^{n-1}f(k,n)$ as a definition by itself. For example, you don't need to know the Gamma function to deal with $\sum_{k=0}^{n-1} \log(k+1)$. 
(by the way, the author advocates using $\sum_{k=0}^{n-1}f(k)$ instead of $\sum_{k=1}^n f(k)$ because the former is elegant mathematically) 
