List of summation formulas in analysis

Question

I know there are various summation formulas exist in analysis:

1.Euler-Maclaurin summation

2.Abel partial summation

3.Poisson summation

4.Abel-Plana summation

I also know that under some modifications these are inter-equivaltent to one another.

My question:

Are there other well known summation formula than above which have wide range of applications.

Answer 1

The following list is by no means complete, it's just meant to be a mere addition to pique your interests:

$(1)$ Borel summation: (other related methods in the linked wiki page are it's generalizations for different kinds of functions according to the requirement at hand)
This summation method for divergent series finds use in the moment problems, where, if you are given a sequence of numbers $\{\beta_i\}_{i\ge 1}$, you have to determine if there exists a measure having these particular moments. For simplicity you can consider the measure to be a probability distribution and $\beta_i$-s to be the moments of the distribution. There are a few related theorems that tell us about the existence and uniqueness and such distribution for a given moment sequence.

$(2)$ Cesaro summation: This method is used to assign a value to some series like the famous Grandi's series $(1-1+1-1+1+\cdots)$ by defining a Cesaro sum, which is an example of a series having a Cesaro sum but is not summable in the usual sense (i.e. convergence of partial sums). However, the converse that if a series converges in the usual sense, it's sum is equal to the Cesaro sum is sometimes guised as the following theorem $$a_n,a\in\Bbb{R},\ \lim_{n\to\infty} a_n =a \implies \lim_{n\to\infty} \frac{\sum_{i=1}^n a_n}n=a$$ The method originally introduced by Ernesto Cesaro was for matrices and the method for real numbers is related to Holder summation. The concept of the Cesaro mean (the term on the right hand side above) was improved by Riesz who introduced the Riesz mean as another method of checking summability of a series.

$(3)$ Ramanujan summation: This should have appeared at the beginning of the list owing to how mysterious Ramanujan and his methods were, but that is the exact reason why it's here - I don't know much about it except it's name. However, this method has it's well-deserved fame. Should you be interested, you can find soft copies of works on Ramanujan and his methods online (by Bruce C. Berndt), my recent favourite source of reading being Paramanand Singh's mathematics blog as a fairly comprehensive way of building things up.