On Entire Functions with Algebraic Derivatives at Certain Algebraic Points

Abstract

plete. In this paper we establish certain lower bounds for the order and type of transcendental entire functions with this property. We then note that these results can be generalized to the case of transcendental entire functions all of whose derivatives are algebraic at a finite number of algebraic points, provided these derivatives satisfy certain conditions, similar to, but slightly less restrictive than those satisfied by Siegel's E functions at the origin [5]. Thus this study gives rise to a new method for investigating the transcendentality of the values of certain entire functions at algebraic points, in particular it yields a new proof of the transcendentality of ea where a is a non-zero algebraic number. This method puts fewer restrictions on the entire functions investigated, since unlike the E functions their order is only assumed finite but not equal to one, and they are not assumed to satisfy differential equations. However the results obtained are less complete and it will usually be difficult to investigate the algebraic character of all the derivatives of a function at certain points without the help of a differential equation.'