On q-summation and confluence

TL;DR

This paper explores q-summation in irregular singular analytic q-difference equations, showing how Borel sums of divergent solutions relate to meromorphic solutions for q in (0,1) and clarifying relations among different q-Borel sums for q in (1,+∞).

Contribution

It provides a uniform approximation of divergent solutions by meromorphic solutions for q in (0,1) and clarifies the relations among multiple q-Borel sums for q in (1,+∞).

Findings

Borel sum of divergent solutions can be approximated by meromorphic solutions for q in (0,1)

Relations among different q-Borel sums are clarified for q in (1,+∞)

The work bridges solutions of differential equations and q-difference equations in different regimes.

Abstract

This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of such a q-difference equation. In the second part, we work under the assumption q\in ]1,+\infty[. In this case, at least four different q-Borel sums of a divergent solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.