On the irrationality measure for certain series

Let K be either the rational number field Bbb Q{Bbb Q} or an imaginary quadratic field. We give irrationality results for the number q = ?_n=1^￥rⁿ/(qⁿ-r^l)theta=sum_{n=1}^{infty}{r^n}/(q^n-r^l), where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and 1 ￡ l ? Bbb Z1le lin{Bbb Z} with q ⁿ ≠ r ^l (n ≥ 1).