Local monodromy formula of Hadamard products

Ricardo Pérez-Marco (1, 2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 CNRS - Centre National de la Recherche Scientifique

We find an explicit general formula for the iterated local monodromy of singularities of the Hadamard product of functions with integrable singularities. The formula implies the invariance by Hadamard product of the class of functions with integrable singularities with recurrent monodromies. In particular, it implies the recurrence of the local monodromy of functions with finite Hadamard grade as defined by Allouche and Mendès-France. We give other examples of natural classes of functions with recurrent monodromies, functions with algebro-logarithmic singularities, and more generally with polylogarithm monodromies. We sketch applications to elliptic integrals, hypergeometric functions, and to fractional integration.

Domaines: Analyse classique [math.CA]; Variables complexes [math.CV]; Théorie des nombres [math.NT]

Ricardo Pérez-Marco. Local monodromy formula of Hadamard products. 2020. ⟨hal-03017278⟩