Some theorems on diophantine approximation

Abstract

Introduction.The study of the values at rational points of transcendental functions defined by linear differential equations with coefficients in Q [z] (2) can be traced back to Hurwitz [1] who showed that if , .,where « is a positive integer, b is an integer, and b\a is not a negative integer, then for all nonzero z in Q(( -1)1/2) the number y'(z)jy(z) is not in g(( -1)1/2).Ratner [2] proved further results.Then Hurwitz [3] generalized his previous results to show that if nZ) ,+g(0) 1! +g(0)-g(l)2!+ where f(z) and g(z) are in Q[z\, neither f(z) nor g(z) has a nonnegative integral zero, and degree (/(z)) < degree (g(z)) = r, then for all nonzero z in the imaginary quadratic field Q(( -n)1'2) two of the numbers y(z),y(l)(z),---,yir\z) have a ratio which is not in Q(( -n)112).Perron [4], Popken [5], C. L. Siegel [6], and K. Mahler [7] have obtained important results in this area.In this paper we shall use a generalization of the method which was developed by Mahler [7] to study the approximation of the logarithms of algebraic numbers by rational and algebraic numbers.Definition.Let K denote the field Q(( -n)i/2) for some nonnegative integer «.Definition.For any monic 0(z) in AT[z] of degree k > 0 and such that 6(z) has no positive integral zeros we define the entire function oo d f(z)= £ ---.