Resurgence of the Euler-Maclaurin summation formula
The Euler-MacLaurin summation formula relates summation to integration as follows: \[ \sum _{k=1}^Nf(k/N)=N\int _0^1f(s)ds+(1/2)(f(1)-f(0))+R(f,N) \] where the remainder term \(R(f,N)\) has an asymptotic expansion given in terms of Bernoulli numbers and values of the derivatives of \(f\) at \(0\) and \(1\). For a typical analytic function this expansion 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) proved that the asymptotic expansion is a Borel summable series, and gave an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, the authors give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, their summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. Finally, they give two applications of the results. One concerns the construction of solutions of linear difference equations with a small parameter. Another concerns resurgence of 1-dimensional sums of quantum factorials, that are associated to knotted 3-dimensional objects.
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