The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series,...

The paper is concerned with the resurgence of the Kontsevich-Zagier series
$$f\left(q\right)=\sum _{n=0}^{\infty}(1-q)\cdots (1-{q}^{n})$$
We give an explicit formula for the Borel transform of the power series when $q={e}^{1/x}$ from which its analytic continuation, its singularities (all on the positive real axis) and the local monodromy can be manifestly determined. We also give two formulas (one involving the Dedekind eta function, and another involving the complex error function) for the right, left and median summation of the Borel transform....

For the hypoelliptic differential operators $P={\partial}_{x}^{2}+{\left({x}^{k}{\partial}_{y}-{x}^{l}{\partial}_{t}\right)}^{2}$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

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