Reference request to study Borel summation Could someone recommend sources to learn about Borel summation procedure?
Books, articles or reviews? I have a background in basic analysis.
 A: The best reference I can think of is a nice book published by OUP:

Bruce Shawyer, and Bruce Watson.
  Borel's methods of summability. Theory and applications. 
  Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1994. xii+242 pp. ISBN: 0-19-853585-6
  MR1320266 (96e:40001).

The book is very clear, it provides a unifying theory for different summation methods, with emphasis on Borel's (in particular, the treatment highlights common themes for summation methods involving sequences, functions, and integrals). It also includes extensive references, and applications.
Here is the list of chapters, to give you an idea of the book's scope:


*

*Historical overview.

*Summability methods in general.

*Borel's method of summability.

*Relations with the family of circle methods.

*Generalizations of Borel's methods.

*Abelian theorems.

*Tauberian theorems I.

*Tauberian theorems II.

*Relationships with other methods.

*Applications of Borel's methods.

*References.

