A note on -operators of order 2
Tom 170 / 2022
Colloquium Mathematicum 170 (2022), 321-340
MSC: Primary 33E20; Secondary 34M15, 34M35, 11J91.
DOI: 10.4064/cm8600-3-2022
Opublikowany online: 4 July 2022
Streszczenie
It is known that G-functions solving a linear differential equation of order 1 with coefficients in \overline {\mathbb Q}(z) are algebraic (and 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 solving an inhomogeneous equation of order 1 with coefficients in \overline {\mathbb Q}(z), as well as that of a G-function f of differential order 2 over \overline {\mathbb Q}(z) and such that f and f’ are algebraically dependent over \mathbb C(z). Our results apply more generally to holonomic Nilsson–Gevrey arithmetic series of order 0 that encompass G-functions.
