Note on the sums of powers of consecutive $q$-integers
  In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open.   Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers. 
Y. Simsek, D. Kim, T. Kim and S. H. Rim