On the diophantine approximation of values of functions satisfying certain linear q-difference equations

Abstract

Lower bounds are obtained on the simultaneous diophantine approximation of some values of certain functions satisfying q difference equations. In particular, a “best possible” type of result is obtained for the individual diophantine approximation of the numbers 1+ zq −1 1+ zq −2 1+ … zq −n 1+ , where q denotes an integer, z denotes a rational number, | q| > 1, and | z| > 0. For z = 1, the above number equals ∏ n=0 ∞ (1−q −(5n+2))(1−q −(5n+1)) −1(1−q −(5n+3))(1−q −(5n+4)) −1