... studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory( see [17-19]). In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of ... Republic of Korea3Department of WirelessCommunications Engineering, Kwangwoon University, Seoul 139-701, Republic of KoreaCompeting interests The authors declare that they have no competing interests.Received: ... medium,provided the original work is properly cited.Therefore, by (13), we obtain the following theorem.Theorem 3. For n Î ℤ+, we haveβn,ξ−1,q−1(1 − x)=ξqn(−1)nβn,ξ,q(x).From Theorem...