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arXiv · 2107.03536

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

Changgui Zhang
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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.

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Changgui Zhang. 2021-06-29. On the positive powers of $q$-analogs of Euler series. https://arxiv.org/abs/2107.03536

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