2009/12/18 by Jean Écalle, Jean Ecalle, Ecalle, Jean +2
Mathematics · #30F99 #34D99 #57N10 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #math.CA #msc:30F99 #msc:34D99 #msc:57N10
paper · pdf · doi:10.48550/arxiv.0912.3687
A number of typos have been rectified and chapter 6 has been expanded
openalex publication_date 2009/12/18 · arxiv created 2010/03/01 · arxiv updated 2010/03/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
The present paper is devoted to power series of SP type, i.e. with coefficients that are syntactically sum-product combinations. Apart from their applications to analytic knot theory and the so-called "Volume Conjecture", SP-series are interesting in their own right, on at least four counts: (i) they generate quite distinctive resurgence algebras (ii) they are one of those relatively rare instances when the resurgence properties have to be derived directly from the Taylor coefficients (iii) some of them produce singularities that unexpectedly verify finite-order differential equations (iv) all of them are best handled with the help of two remarkable, infinite-order integral-differential transforms, "mir" and "nir".