2021/06/10 by Fischler, Stéphane, Rivoal, Tanguy · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.05572
It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in ℚ(z), are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in ℚ(z), as well as that of a G-function f of differential order 2 over ℚ(z), and such that f and f' are algebraically dependent over ℂ(z). Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.