On the positive powers of 𝑞-analogs of Euler series

Abstract

The most simple and famous divergent power series coming from ODE may be the so-called Euler series ∑ n ≥ 0 ( − 1 ) n n ! x n + 1 \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 (see \cite{Ba} or \cite{Ma}). By considering a family of linear q q -difference operators associated with a given first order non-homogenous q q -difference equation, it will be shown that the summability order of q q -analoguous counterparties of Euler series depends upon of the degree of power under consideration.