On the positive powers of $q$-analogs of Euler series

TL;DR

This paper investigates the summability properties of $q$-analogs of Euler series, revealing how the summability order varies with the power degree through $q$-difference operators.

Contribution

It introduces a family of $q$-difference operators to analyze the summability of $q$-Euler series and shows the dependence of summability order on the power degree.

Findings

Summability order depends on the power degree of the $q$-Euler series.

$q$-analogs of Euler series are Borel-summable except on the negative real axis.

The approach links $q$-difference equations with summability properties.

Abstract

The most simple and famous divergent power series coming from ODE may be the so-called Euler series $\sum_{n\ge 0}(-1)^n\,n!\,x^{n+1}$, that, as well as all its positive powers, is Borel-summable in any direction excepted the negative real half-axis. By considering a family of linear $q$-difference operators associated with a given first order non-homogenous $q$-difference equation, it will be shown that the summability order of $q$-analoguous counterparties of Euler series depends upon of the degree of power under consideration.