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 ${\sum }_{n\ge 0}(-1{)}^{n}n!x^{n+1}$, 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.