Power series with sum-product Taylor coefficients and their resurgence algebra

TL;DR

This paper studies a special class of power series with sum-product coefficients, exploring their unique resurgence properties, algebraic structures, and the use of novel integral transforms, with implications for knot theory and differential equations.

Contribution

It introduces the analysis of SP-type power series, revealing their resurgence algebras, differential equations, and the application of new integral transforms, advancing understanding of their mathematical structure.

Findings

SP-series generate distinctive resurgence algebras

Some SP-series satisfy finite-order differential equations

Two integral transforms, 'mir' and 'nir', are key tools for analysis

Abstract

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".