  Carlitz has introduced q-analogues of the Bernoulli numbers around 1950. We
obtain a representation of these q-Bernoulli numbers (and some shifted version)
as moments of some orthogonal polynomials. This also gives factorisations of
Hankel determinants of q-Bernoulli numbers, and continued fractions for their
generating series. Some of these results are q-analogues of known results for
Bernoulli numbers, but some are specific to the q-Bernoulli setting.
