On the irrationality measure for certain series

Let K be either the rational number field or an imaginary quadratic field. We give irrationality results for the number , where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and with q ⁿ ≠ r ^l (n ≥ 1). Authors’ addresses: Kenji Amano, NS solutions Corporation, 2-27-1 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan; Yohei Tachiya, Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan