Complex Variables 29:41-44 (1996)

© 1996 OPA (Overseas Publishers Association) Amsterdam B.V.

AMS 1991: 34E05, 41A60, 12H05, 12J25, 26A12

Abstract: We prove that a formal power series in 1/x, whose coefficients are in a field extension of Q and are algebraically independent over Q, is differentially transcendental (i.e. not differentially algebraic) over this field extension. This is stated without proof in [M. Boshernitzan, New "Orders of Infinity", J. d'Analyse Math 41:130-167 (1982)]. This result provides a source of functions analytic at that are not differentially algebraic over R. Such functions are of particular interest, because their germs belong to Hardy fields, but not to the class E of [M. Boshernitzan, An Extension of Hardy's Class L of "Orders of Infinity", J. d'Analyse Math 39:235-255 (1981)] - the intersection of all maximal Hardy fields.

@article{dg:diftran,author={Gokhman, D.},
title={Differentially transcendental formal power series},
journal={Complex Variables},volume={29},pages={41--44},year={1996}}