Resurgence-summability of ramified formal series depending of a parameter and solutions of linear differential equations
The author investigates the Borel-resummable and resurgent properties of formal power series, solutions of linear differential equations of the form \[ Q_n(x,b)d^2\Phi (x)/dx^2=P_m(x,a)\Phi (x), \] where \[ P_m=x^m+a_1x^{m-1}+\cdots +a_m,\quad Q_n=x^n+b_1x^{n-1}+\cdots +b_n. \] In particular, he studies the analytic dependence of the associated Borel sum in the parameters of the equations. He also analyses the whole resurgent structure associated with these formal power series thanks to the tool of microfunctions and the so-called alien derivatives. This analysis allows to extend some results due to \textit{Y. Sibuya} in [Global theory of a second order linear differential equation with a polynomial coefficient. Amsterdam-Oxford: North-Holland Publishing Company; New York: American Elsevier Publishing Company (1975; Zbl 0322.34006)], and to illustrate the powerful alien calculus of J. Ecalle.
scientific article; zbMATH DE number 5799093
scientific article