The next meeting of the MAX team seminar will be on Tuesday, May 24. The meeting is dedicated to the Computer Mathematics research group. We will welcome Marc Mezzarobba (LIX) for his talk on Asymptotic Expansions with Error Bounds for Solutions of Linear Recurrences.

Abstract: When a sequence (u(n)) of real or complex numbers satisfies a linear recurrence relation with polynomial coefficients, it is usually routine to determine its asymptotic behavior for large n. Well-known algorithms and readily available implementations are often able, given a recurrence satisfied by (u(n)) and some additional information on the particular solution, to compute an explicit asymptotic expansion

u(n) = a₀(n) + a₁(n) +  ·  ·  ·  + a_r − 1(n) + O(b(n)), n → ∞

up to any desired order r. If, however, one wishes to prove an inequality satisfied by u(n) for all n, a big-Oh error term is not sufficient and an explicit error bound on the difference between the sequence and its asymptotic approximation is required. In this talk, I will present a practical algorithm for computing such bounds, under the assumption that the generating series of the sequence (u(n)) is solution of a differential equation with regular (Fuchsian) dominant singularities. I will show how it can be used to verify the positivity of a certain explicit sequence, which T. Yu and J. Chen recently proved to imply uniqueness of the surface of genus one in ℝ³ of minimal bending energy when the isoperimetric ratio is prescribed. Based on joint work with Ruiwen Dong and Stephen Melczer